G8_Set_Lesson_All

Set Overview

Browse the course units below.

Set

Introduction

Set

Representation

Set

Cardinality

Set

Types

Set

Subset

Set

BLE Questions

MOOC Test

Set

Set

Set (समुह) आधुनिक गणितको एउटा आधारभुत आवधारण हो। यो अवधारणा गणितका लगभग प्रत्येक शाखामा प्रयोग हुन्छ। Set (समुह) को बारेमा जर्मन गणितज्ञ जर्ज कान्टर (१८४५-१९१८) ले चर्चा गरेका थिए। उनले पहिलो पटक "त्रिकोणमितीय श्रेणीका समस्याहरू" समाधान गर्ने क्रममा Set (समुह) प्रयोग गरेका थिए।

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समूह भनेको परिभाषित गर्न सकिने वा निश्चित वस्तुहरूको सङ्ग्रह हो, जसलाई अंग्रेजी वर्णमालाका ठूलो अक्षरले जनाइन्छ, जस्तै \(A, B, C \ldots \)।
उदाहरणको लागी
\(A = \{a, e, i, o, u\}\) (1): The set of vowels
Set is a well defined collection of objects.
समुह भनेको राम्रोसँग परिभाषित गर्न सकिने वस्तुहरूको सङ्ग्रह हो।

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How do we know if a set is well defined?

समूहमा "परिभाषित" भन्नाले सदस्यहरूलाई \(\in\) वा \(\notin\) प्रयोग गरि व्याख्या गर्न सकिने कुरालाई जनाउदछ। जस्तै,
\(A = \{a, e, i, o, u\}\) (1): The set of vowels
Here, (1) मा दिइएको समूह \(A\) परिभाषित छ किनभने हामी भन्न सक्छौं की
\(a \in A , b \notin A\)
यँहा, \(\in\) को अर्थ "अन्तर्गत पर्छ" वा "सदस्य हो" भन्ने हुन्छ। यदि कुनै पनि कुरा दिएको समुहको सदस्य भएमा, हामी \(\in\) प्रयोग गर्छौं। र \(\notin\) को अर्थ "अन्तर्गत पर्दैन" वा "सदस्य होईन" भन्ने हुन्छ। यदि कुनै पनि कुरा दिएको समुहको सदस्य नभएमा, हामी \(\notin\) प्रयोग गर्छौं। जस्तै-
समुह \(A\) मा \(a\) पर्दछ, त्यसैले \(a \in A\)
समुह \(A\) मा \(b\) पर्दैन त्यसैले \(b \notin A\)

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Member of set

समूहका सदस्यहरुलाई member भनिन्छ, जसलाई मझौला कोष्ठ \(\{\cdots\}\) भित्र राखिन्छ। समूहका सदस्यहरू भौतिक वस्तुहरू जस्तै किताब, कलम, व्यक्ति वा धारणात्मक वस्तुहरू जस्तै सङ्ख्या, विन्दु वा अन्य प्रकारका बस्तुहरु पनि हुन सक्छ। समूहका सदस्यहरू अंग्रेजी वर्णमालाका अक्षरहरू छन भने साना अक्षरहरू लेखेर जनाइन्छ ।

Symbol	Name	Example	Explanation
\(\{ \}\)	Set	\(A = \{a,e,i,o,u\}\)	The set of vowels
\(\in\)	Membership	\(a \in A, e \in A, i \in A, o\in A,u \in A\)	The symbol \(\in\) denotes membership
\(\notin\)	Non-membership	\(5 \notin A, b \notin A\)	The symbol \(\notin\) denotes non-membership

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The marks of a few students of class 8 in a school are given below.

Bidhi- 45

Bidhan- 43

Ram- 52

Shyam- 49

Pemba- 41

Najir- 46

Min- 51

Najma- 48

1. Can you make a set of 'talent students'? Give reason.
2. Can you make a set of students with marks 'more than 45'? Give reason.
3. Can you make a set of students with marks 'less than 43'? Give reason.

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Let's say

\(A = \{Fe, Fo\}\) and \(B = \left\{1.3,\ \pi,\ \sqrt[3]{2},\ \frac{1}{3},\ 3.33\cdots\right\}\)

1. Is the set \(A\) well defined? Give reason.
2. Is the set \(B\) well defined? Give reason.

1. Yes, the set \(A = \{Fe, Fo\}\) is well defined because the terms “Fe” and “Fo” are clearly specified.
2. Yes, the set \(A = \{1.3,\ \pi,\ \sqrt[3]{2},\ \frac{1}{3},\ 3.33\cdots\}\) is well defined because the elements are clearly specified.

For a set to be well defined, its elements must be clearly identifiable and there should be no ambiguity to use membership element in the set.

\(Fe \in A\) and \(Fo \in A\)

\(1.3 \in B\) and \(4 \in B\) and so on.

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तलको तालिकामा भएका समुहहरु "परिभाषित" छ वा छैन थाहा पाउनको लागी विचार गरि
\(True-T\) वा
\(False-F\) वा
\(\text{Not applicable}\)-NA
लेख्नुहोस र
समुहहरु ``परिभाषित" भएमा ``Yes" वा ``परिभाषित" नभएमा ``No" लेख्नुहोस।

SN	Set	Statement	T / F / NA	Statement	T / F / NA	परिभाषित
1	\( A= \{1,2,3,4,5\} \)	\(1 \in A\)	--TFNA	\(2 \notin A\)	--TFNA	--YesNo
2	\( B= \{\text{red, blue, green, yellow}\} \)	\(\text{red} \in B\)	--TFNA	\(\text{pink} \notin B\)	--TFNA	--YesNo
3	\( C = \{x \mid x \text{ is an even number less than } 10\} \)	\(3 \in C\)	--TFNA	\(5 \notin C\)	--TFNA	--YesNo
4	\( D = \{\text{all vowels in English alphabet}\} \)	\(a \in D\)	--TFNA	\(p \notin D\)	--TFNA	--YesNo
5	\( E = \{x \mid x \text{ is a prime number less than } 20\} \)	\(18 \in E\)	--TFNA	\(19 \notin E\)	--TFNA	--YesNo
6	\( F = \{\text{smart students in the class}\} \)	\(\text{Gita} \in F\)	--TFNA	\(\text{Ram} \notin F\)	--TFNA	--YesNo
7	\( G = \{\text{tall students in the school}\} \)	\(\text{Mohan} \in G\)	--TFNA	\(\text{Rani} \notin G\)	--TFNA	--YesNo
8	\( H = \{\text{good books}\} \)	\(\text{Math} \in H\)	--TFNA	\(\text{Poem} \notin H\)	--TFNA	--YesNo
9	\( I = \{\text{beautiful flowers}\} \)	\(\text{Rose} \in I\)	--TFNA	\(\text{Lily} \notin I\)	--TFNA	--YesNo
10	\( J = \{\text{best football players}\} \)	\(\text{Hari} \in J\)	--TFNA	\(\text{Ram} \notin J\)	--TFNA	--YesNo

Exercise

1. If \(A = \{2, 4, 6, 8, \cdots\}\), write 'true' or 'false'.

1. \(6 \in A\) -- true false
2. \(12 \in A\) -- true false
3. \(5 \in A\) -- true false
4. \(10 \notin A\) -- true false
5. \(15 \notin A\) -- true false
6. \(18 \notin A\) -- true false
7. Is the set \(A\) well defined? --- Yes No

2. Use (✓) for well-defined collections, and (✗) for others.

1. A collection of Nepali movies released in 2081 B.S. -- ✓ ✗
2. A collection of favourite Nepali movies released in 2081 B.S. -- ✓ ✗
3. A collection of smaller prime numbers less than 10. -- ✓ ✗
4. A collection of prime numbers less than 10. -- ✓ ✗

3. Let's take a collection of any three "high mountains of Nepal" and answer:

1. Is it a well-defined collection? Give reason. --- Yes No
2. Is it a set? Give reason. --- Yes No
3. Express it as a well-defined collection and list members.

4. How do you know if a set is well defined?

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What are different ways to representation a Set?

समूहलाई साधारणतया चार प्रमुख तरिकाले प्रस्तुत गर्न सकिन्छ जस्तै (1) सूचीको रूपमा -roster form, (2) सङ्केतको रुपमा -set-builder form, (3) वर्णनात्मक रूपमा -descriptive form, र (4) भेन-चित्रको रुपमा -Venn-diagram form।

Method	Example	Explanation
Description	\(A\) is a set of multiples of 3 less than 15.	described by words.
Listing (or roster)	\(A = \{3, 6, 9, 12\}\)	elements are listed inside \(\{\}\).
Set-builder (or rule)	\(A = \{x : x \in \text{multiples of } 3, x < 15\}\)	variable \(x\) describe the properties

1. roster form : \(A = \{a, e, i, o, u\}\)
   The roster form (also called the "listing method"), explicitly lists all the elements of the set within curly braces \(\{\}\). Each element is written only once, and the order of elements does not matter.
   For example, \(\{a,e,i,o,u\}\) is the same as \(\{u,o,i,e,a\}\).
2. set builder form : \(A = \{x: x \text{ is a vowel}\}\)
   The set builder form describes a set by specifying a property that all elements of the set must satisfy. It is written as \(\{x: \text{condition on } x\}\), which is read as "the set of all \(x\) such that the condition on \(x\) is true."
   This method is useful when listing all elements is impractical or impossible.
3. descriptive form : \(A = \{\text{set of vowels}\}\)
   The descriptive form uses plain language to describe the set. Instead of listing elements or using mathematical notation, the set is described in words.
   
   Here, the set \(A\) is described as "the set of vowels." This is a simple and intuitive way to convey the idea of the set without using formal mathematical notation. This method is often used for quick explanations or informal contexts.
4. Venn diagram form
   A Venn diagram is a visual representation of a set using circles or oval shapes, and the elements of the set can be shown inside the region. Venn diagrams are useful for visualizing relationships between sets, such as unions, intersections, and complements. They are particularly useful in problems involving multiple sets.
   भेनचित्रको प्रयोग सन् 1880 मा गणितज्ञ John Venn ले गरेका थिए । उनकै नामबाट यस चित्रलाई भेन चित्र भनिएको हो ।

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Exercise

तल दिइएको सङ्कलित संख्याका आधारमा प्रश्नहरूको उत्तर दिनुहोस्।
Answer the following questions based on the given collected numbers:
\(A=\{2, 3, 5, 7, 11, 13, 17, 19\}\)

1. दिइएको सङ्कलन परिभाषित (well-defined) छ कि छैन, कारणसहित उल्लेख गर्नुहोस्।
2. दिइएको सङ्कलनलाई समूह संकेत (set notation) मा लेख्नुहोस्।
3. दिइएको सङ्कलनलाई सेट-बिल्डर (set-builder) रूपमा लेख्नुहोस्।
   Write the given collection in set-builder form.

Cardinality of set

मानौ \(A\) एउटा समुह हो, समुह \( A \) मा भएका सबै सदस्यहरुको सँख्यालाई cardinality (or cardinal number ) of \( A \) भनिन्छ ।
यसलाई \(n(A) \) अथवा \( |A|\) अथवा \( \text{Card}(A)\) संकेतले जनाईन्छ।
जस्तै
If \( A = \{x: x < 4, x \in \mathbb{W} \} \) then \( A =\{0, 1, 2, 3\}\) and \(n (A) = 4\).
सामान्य रूपमा, समुहको cardinality ले तलका गुणहरू जनाउदछ।

1. प्रत्येक समूहको एक मात्र cardinal सङ्ख्या हुन्छ।
2. दुई वटा equivalent समूहको cardinal सङ्ख्या एउटै हुन्छ।
3. गन्ती गर्न सकिने समूहको cardinal सङ्ख्या \(n\) हुन्छ।
4. गणनायोग्य (denumerable) समूहको cardinal सङ्ख्या ‘एलेफ नल’ \(\aleph_0\) हुन्छ।
5. गन्ती गर्न नसकिने (uncountable) समूहको cardinal सङ्ख्या \(c\) हुन्छ।

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Find the cardinality of the following sets.

1. \(A = \{1, 4, 9, 16\}\)
   
   
2. \(B = \{1, 2, 3, 4, 5, \ldots, 15\}\)
   
   
3. \(C = \{2, 4, 6, 8\}\)
   
   
4. \(D = \{0, 1, 2, 3, 4, 5, \ldots, 10\}\)
   
   
5. \(E = \{1, 2, 4, 5, 10, 20\}\)
   
   

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A

What is the cardinal number of set A?

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कक्षाकोठामा भएका बिद्यार्थी वा बस्तु लाई प्रयोग गरि बनाईएका फरक फरक समुहहरुको cardinality सँख्या पत्ता लगाउन लगाउनुहोस।

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Let \(P = \{x : x \text{ is a natural number, } x \leq 20\}\), \(Q = \{x : x \text{ is a factor of } 20\}\) and \(R = \{z : z \text{ is a multiple of } 3, z ≤ 20\}\), then.

1. List the elements of the sets \(P,Q,R\)
2. List the common elements of the sets \(P,Q\)
3. Identify the cardinality of common elements of the sets \(P,Q\)

1. The elements of the sets \(P, Q, R\) are
   \(P = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}\)
   \(Q = \{1, 2, 4, 5, 10, 20\}\)
   \(R = \{3, 6, 9, 12, 15, 18\}\)
2. The common elements of the sets \(P\) and \(Q\) is
   \( \{1, 2, 4, 5, 10, 20\} \)
3. The cardinality of common elements of the sets \(P, Q\) is
   \(6\)

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The factor-cards given below.

1 2 3 4 6 12

1 2 3 6 9 18

1. Write the set of factors of 12 (\(F_{12}\)) in listing method.
2. Write the set of factors of 18 (\(F_{18}\)) in listing method.
3. Write the set of factors of 18 (\(F_{18}\)) in set-builder method.
4. List the common elements of \(F_{12}\) and \(F_{18}\) in a separate set \(P\) and write its cardinal number.

1. The set of factors of 12 (\(F_{12}\)) in listing method is
   \(F_{12} = \{1, 2, 3, 4, 6, 12\}\)
2. The set of factors of 18 (\(F_{18}\)) in listing method is
   \(F_{18} = \{1, 2, 3, 6, 9, 18\}\)
3. The set of factors of 18 (\(F_{18}\)) in set-builder method is
   \(F_{18} = \{x : x \text{ is a positive integer and } x \text{ divides } 18\}\)
   or\(F_{18} = \{x : x \mid 18,\ x \in \mathbb{N}\}\)
4. The common elements of \(F_{12}\) and \(F_{18}\) in a separate set \(P\) is
   \(P = \{1, 2, 3, 6\}\)
   The cardinal number of \(P\) is
   \(n(P) = 4\)

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The multiple-cards given below show first five multiples of 4 and 7.

4 8 12 4 16 20

7 14 21 28 35

1. Write the set of the first five multiples of 4 (\(M_4\)) and 7 (\(M_7\)) in listing method.
2. Write the sets \(M_4\) and \(M_7\) in set-builder method.
3. Write a set \(A\) taking the common elements of \(M_4\) and \(M_7\).
4. What is cardinality of set \(A\)?

1. The set of the first five multiples of 4 (\(M_4\)) and 7 (\(M_7\)) in listing method are
   \(M_4 = \{4, 8, 12, 16, 20\}\)
   \(M_7 = \{7, 14, 21, 28, 35\}\)
   
2. The sets \(M_4\) and \(M_7\) in set-builder method are
   \(M_4 = \{x : x = 4n,\ n \in \mathbb{N},\ 1 \leq n \leq 5\}\)
   \(M_7 = \{x : x = 7n,\ n \in \mathbb{N},\ 1 \leq n \leq 5\}\)
   
3. The set \(A\) taking the common elements of \(M_4\) and \(M_7\) is
   \(A = \{\}\) or \(A = \varnothing\)
   
4. The cardinality of set A is 0.

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Types of sets

समुहका प्रकारहरु निम्नानुसार छन।

1. Empty set
   कुनै पनि सदस्य (तत्व) नभएको समूहलाई रिक्त (empty) वा शून्य (null) वा रित्तो (void) समूह भनिन्छ, र यसलाई \(\{ \}\) वा \(\phi\) ले जनाइन्छ।जस्तै
   \(A = \{ x: x \text{ is a natural number less than } 1\} = \phi \)
2. Singleton set
   एउटा मात्र सदस्य (तत्व) भएको समूहलाई एकल समूह भनिन्छ।जस्तै
   \(A = \{ x: x \text{ is even prime}\} = \{2\} \)
3. Finite set
   गन्ती गर्न सकिने सीमित संख्याहरु भएको समूहमा लाई finite set भनिन्छ।जस्तै
   \(A = \{ \text{ whole numbers less than } 10\} = \{ 0,1,2,3,4,5,6,7,8,9\} \)
4. Infinite set
   गन्ती गर्न नसकिने असीमित संख्याहरु भएको समूहमा लाई infinite set भनिन्छ।जस्तै
   \(B = \{ x: x \in \mathbb{W}, x = 2n\} = \{ 0, 2, 4, 6, \ldots\} \)

Overlapping and Disjoint Set

सार्क राष्ट्रहरू अन्तर्गत पर्ने देशहरूमध्ये केही राष्ट्रहरू समुद्रसँग जोडिएका छन् भने केही राष्ट्रहरू भूपरिवेष्टित छन। यसलाई समूहमा बिभिन्न तरिकाले जनाउन सकिन्छ । जस्तै
A={समुद्रसँग जोडिएका राष्ट्रहरू}
B={भूपरिवेष्टित राष्ट्रहरू}
C={भारतसँग सिमा जोडिएका राष्ट्रहरू}
दिइएको नक्शा अध्ययन गरी \(A,B\) र \(A,C\) लाई समूहमा कसरी जनाउन सकिन्छ बिचार गर्नुहोस? यि दुबैको भेन चित्र Overlapping वा Disjoint के हुन्छ? तलको भेन चित्रमा तयार गर्नुहोस।

Drag and Drop Quiz: Q1

Look at the map above and, drag each country to all correct categories it belongs to. Please note that some countries belong to more than one group!

A: Sea Access

B: Landlocked

C: Borders India

Score: 0 / 12

सार्क राष्ट्रहरू अन्तर्गत पर्ने देशहरूमध्ये केही राष्ट्रहरू समुद्रसँग जोडिएका छन् भने केही राष्ट्रहरू भूपरिवेष्टित छन। यसलाई समूहमा बिभिन्न तरिकाले जनाउन सकिन्छ । जस्तै
A={समुद्रसँग जोडिएका राष्ट्रहरू}
B={भूपरिवेष्टित राष्ट्रहरू}
C={भारतसँग सिमा जोडिएका राष्ट्रहरू}
यि समूहको भेन चित्र Overlapping वा Disjoint के हुन्छ?

In this activity,
the sets A and B are
Overlapping Disjoint
the sets A and C are
Overlapping Disjoint

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Overlapping Set

कुनै दुई वा दुईभन्दा बढी समूहमा कम्तीमा एउटा साझा सदस्य र कम्तीमा एउटा फरक सदस्य छ भने त्यस्ता समूहलाई खप्टिएका समूह (Overlapping Set) भनिन्छ ।
जस्तै,
यदि \( A = \{1,2,3\} \) र \( B = \{3,4,5\} \) छन् भने \( A \) र \( B \) खप्टिएको समुह हो किनभने \(\{3\}\) दुबैमा साझा सदस्य हो। चित्र [overlappingset] हेर्नुहोस्।

Disjoint Set

कुनै दुई वा दुईभन्दा बढी समूहमा कुनै पनि साझा सदस्य छैनन् भने त्यस्ता समूहलाई अलग्गिएका समूह (Disjoint Set) भनिन्छ । जस्तै, यदि \( A = \{1, 2, 3\} \) र \( B = \{4, 5, 6\} \) छन् भने \( A \) र \( B \) परस्पर अलग्गिएका समुह हुन् किनभने तिनीहरूमा कुनै पनि साझा सदस्य छैन। चित्र [disjointset] हेर्नुहोस्।

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दिइएको भेनचित्र अध्ययन गरी समूह M र N लाई सूचीकरण तथा व्याख्या विधिबाट लेख्नुहोस्।

सूचीकरणविधि अनुसार
\(M=\)
\(N=\)
व्याख्या विधि अनुसार
\(M=\)
\(N=\)

सूचीकरणविधि अनुसार
\(M=\{3,,6,9,12,15\}\)
\(N=\{6,12,18,24\}\)
व्याख्या विधि अनुसार
\(M=\) set of the first five positive multiples of 3.
\(N=\) set of the first four positive multiples of 6.

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सर्वव्यापक समूह U का तिनवटा उपसमूहहरु \(P, Q\) र \(R\) छन्। यदि
U ={20 सम्मका 2 को अपवर्त्य हो ।}
P ={10 का गुणनखण्डहरू}
Q ={5 का अपवर्त्य हो ।} हो। र
R ={6 का अपवर्त्य हो ।}
अब, \(P,Q\) तथा \(Q, R\) लाई भेनचित्रमा प्रस्तुत गर्नुहोस्। साथै \(P,Q\) तथा \(Q,R\) को सम्बन्ध पनि उल्लेख गर्नुहोस।

सर्वव्यापक समूह \( U \) र उपसमूहहरू \( P, Q, R \) को परिभाषा अनुसार
\( U = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \)
\( P = \{1, 2, 5, 10\} \cap U = \{2, 10\} \)
\( Q = \{5, 10, 15, 20, \dots\} \cap U = \{10, 20\} \)
\( R = \{6, 12, 18, 24, \dots\} \cap U = \{6, 12, 18\} \)

\( P \) र \( Q \) को सम्बन्ध

Here
\( P \cap Q = \{10\} \)
Therefore, \( P \) र \( Q \) are overlaping sets.

\( Q \) र \( R \) को सम्बन्ध

Here
\( Q \cap R = \emptyset \)
Therefore, \( Q \) र \( R \) are disjoint sets.

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If \( M = \{x : x \text{ is a multiple of 5 less than 40} \} \), list out the elements of the following sets.

1. Set \( N \) is formed by adding 3 to each element of the set \( M \).
2. Set \( P \) is formed by multiplying each element of set \( M \) by 2.
3. Set \( Q \) is formed by collecting all the odd numbers of set \( M \).
4. Set \( R \) is formed by collecting the multiples of 10 from set \( M \).
5. Write the Universal set for the sets given above.

Given that
\( M = \{5, 10, 15, 20, 25, 30, 35\} \)

1. Set \( N \) is formed by adding 3 to each element of \( M \), so
   \( N = \{5+3, 10+3, 15+3, 20+3, 25+3, 30+3, 35+3\} = \{8, 13, 18, 23, 28, 33, 38\} \)
2. Set \( P \)is formed by multiplying each element of \( M \) by 2, so
   \( P = \{2 \times 5, 2 \times 10, 2 \times 15, 2 \times 20, 2 \times 25, 2 \times 30, 2 \times 35\} = \{10, 20, 30, 40, 50, 60, 70\} \)
3. Set \( Q \)is formed by collecting all the odd numbers of \( M \), so
   \( Q = \{5, 15, 25, 35\} \)
4. Set \( R \) is formed by collecting the multiples of 10 from \( M \), so
   \( R = \{10, 20, 30\} \)
5. Universal set \( U \) for the sets \( M, N, P, Q, R \) is
   \( U = M \cup N \cup P \cup Q \cup R \)
   or\( U = \{5, 8, 10, 13, 15, 18, 20, 23, 25, 28, 30, 33, 35, 38, 40, 50, 60, 70\} \)

Identify the overlapping and disjoint sets among sets \( M, N, P, Q, \) and \( R \)

Identify whether each pair of sets is Overlapping (O) or Disjoint (D)

Pair	Overlapping (O)	Disjoint (D)
\( M \) and \( Q \)
\( M \) and \( R \)
\( P \) and \( R \)
\( M \) and \( P \)
\( M \) and \( N \)
\( N \) and \( P \)
\( N \) and \( Q \)
\( N \) and \( R \)
\( Q \) and \( R \)

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If \(U = \{\text{the set of natural numbers up to } 20\}\)

1. \(A = \{\text{the set of even numbers up to } 10\}\)
2. \(B = \{\text{the set of odd numbers up to } 10\}\)
3. \(C = \{\text{the set of prime numbers up to } 10\}\)
4. \(D = \{\text{multiples of 2}\}\).

Specify if the given sets are disjoint or overlap in a table below. Write \(D\) if disjoint, \(O\) if overlapping, \(E\) if equal and \(NA\) if not applicable. Also represent them in separate Venn diagram.

Now, select the correct relationship for each pair:

	A	B	C	D
A	--DOENA	--DOENA	--DOENA	--DOENA
B	--DOENA	--DOENA	--DOENA	--DOENA
C	--DOENA	--DOENA	--DOENA	--DOENA
D	--DOENA	--DOENA	--DOENA	--DOENA

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Prepare a universal set consisting of your family members and do the following work:

1. \(A=\) set of members of your family who likes bread as their breakfast.
2. \(B=\) set of members of your family who likes other breakfast.
3. Represent \(A\) and \(B\) in venn-diagram

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Equal and Equivalent Set

दुई वा बढी समुहहरूमा समान र उही सदस्यहरू छन् भने त्यस्तो समुहहरूलाई बराबर समुह भनिन्छ। जस्तै,
यदि \( A = \{1, 2\}, B = \{2, 1\}, C = \{12\} \) र \( D = \{21\} \) छन् भने
\( A = B \) तर \( C \ne D \) बराबर समुहको संकेत (\(=\)) हो।
दुई वा बढी समुहहरूमा समान सँख्यमा सदस्यहरू छन् भने त्यस्तो समुहहरूलाई समतुल्य समुह भनिन्छ। जस्तै,
यदि \( A = \{1, 2, 3\} \) र \( B = \{p, q, r\} \) छन् भने
\( A \sim B \) समतुल्य समुहको संकेत (\(\sim\)) हो।

Subset

The universal set, denoted by \( U \), is the set that contains all the elements under consideration in a given context. Every other set is a subset of this universal set.

Let the universal set be
\(U = \{ \text{apple}, \text{banana}, \text{mango}, \text{orange}, \text{grape} \}\)
Let set \( A \) be the set of fruits given as
\(A = \{ \text{apple}, \text{mango}, \text{grape} \}\)
Then \(U\) is universal set, and \( A\) is subset.

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Subset

कुनै समुह \( A \) को प्रत्येक सदस्य अर्को समुह \( B \) को पनि सदस्य हो भने, समुह \( A \) लाई समुह \( B \) को उपसमुह भनिन्छ।
यसलाई \( A \subset B \) भनेर लेखिन्छ र “ \( B \) को उपसमुह \( A \) ” भनेर पढिन्छ।
जस्तै: यदि \( A = \{1, 2, 3\}, B = \{3, 4, 5,6\} \) र \( C = \{1, 2, 3, 4, 5\} \) छन् भने
\( A \subset C \) तर \( B \not\subset C \).(चित्र 1 र 2 हेर्नुहोस्)

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In the system of real numbers, with usual notation, the relation between sets are given as
\(\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset\mathbb{Q} \subset \mathbb{R}\).
Define each of the set \(\mathbb{N} , \mathbb{W}, \mathbb{Z} ,\mathbb{Q} , \mathbb{R}\) with one example on each.

1. \(\mathbb{N}\)
   Example: \(5\)
2. \(\mathbb{W}\)
   Example: \(0\)
3. \(\mathbb{Z}\)
   Example: \(-3\)
4. \(\mathbb{Q}\)
   Example: \(\frac{2}{3}\)
5. \(\mathbb{R}\)
   Example: \(\sqrt{2}\)
6. The Venn-diagram of \(\mathbb{N} , \mathbb{W}, \mathbb{Z} ,\mathbb{Q} , \mathbb{R}\) are given below.

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Proper and Improper Subset

In general, there are two types of subsets, they are proper subset and improper subset.

1. उपयुक्त उपसमुह (Proper subset)
   यदि \( A \subset B \) र \( A \ne B \) भने \( A \) लाई \( B \) को उपयुक्त उपसमुह भनिन्छ। यस अवस्थामा, \( B \) लाई \( A \) को सुपर समुह भनिन्छ।
   a set B which contains elements from A but, not all, is called proper subset of A
   If \(B=\{1, 5, 8, 9, 10\}\) , then proper subset of B are
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{1}, {5}, {8}, {9}, {10}	5
   2	{1, 5}, {1, 8}, {1, 9}, {1, 10}, {5, 8}, {5, 9}, {5, 10}, {8, 9}, {8, 10}, {9, 10}	10
   3	{1, 5, 8}, {1, 5, 9}, {1, 5, 10}, {1, 8, 9}, {1, 8, 10}, {1, 9, 10}, {5, 8, 9}, {5, 8, 10}, {5, 9, 10}, {8, 9, 10}	10
   4	{1, 5, 8, 9}, {1, 5, 8, 10}, {1, 5, 9, 10}, {1, 8, 9, 10}, {5, 8, 9, 10}	5
   Total Subsets		32

2. अनुपयुक्त उपसमुह (Improper subset)
   समुहको परिभाषाबाट परम्परागत रूपमा नै, शून्य समुह र समुह आफैंलाई अनुपयुक्त उपसमुह पनि भनिन्छ। त्यसैले, यदि \( A \subset B \) र \( A = B \) भएमा \( A \) लाई \( B \) को अनुपयुक्त उपसमुह भनिन्छ। यसलाई \( A \subseteq B \) ले जनाईन्छ।
   a set B which contains all elements of A, is called improper subset of A
   If\(B=\{1, 5, 8, 9, 10\}\), then improper subset of B is
   \(B=\{1, 5, 8, 9, 10\}\)

NOTE

1. शून्य समुह \( \phi \) प्रत्येक समुहको उपसमुह हो।
2. प्रत्येक समुहको (खाली समुह बाहेक) कम्तीमा दुईवटा उपसमुहहरू हुन्छन्।
3. \( n \) वटा सदस्य भएको समुहको सम्भावित उपसमुहहरु \( 2^n \) वटा हुन्छ, जसको समुहलाई Power Set भनिन्छ।

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If \(A=\{a,e,i,o,u\}\), then

1. find all subsets consisting no element
   \(\{\}\) or \(\phi\)
2. find all subsets consisting 1 element
   \(\{a\}, \{e\}, \{i\}, \{o\}, \{u\}\)
3. find all subsets consisting 2 elements
   \(\{a,e\}, \{a,i\}, \{a,o\}, \{a,u\}, \{e,i\}, \{e,o\}, \{e,u\}, \{i,o\}, \{i,u\}, \{o,u\}\)
4. find all subsets consisting 3 elements
   10 subsets, e.g., \(\{a,e,i\}, \{a,e,o\}, \dots\))
5. find all subsets consisting 4 elements
   5 subsets, e.g., \(\{a,e,i,o\}, \{a,e,i,u\}, \dots\))

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If \(A=\{a\}, B=\{a,b\}, C=\{a,b,c\}, D=\{a,b,c,d\}\), then

1. find all subsets of \(A\)
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{a}	1
   Total Subsets		2

2. find all subsets of \(B\)
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{a}, {b}	2
   2	{a, b}	1
   Total Subsets		4

3. find all subsets of \(C\)
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{a}, {b}, {c}	3
   2	{a, b}, {a, c}, {b, c}	3
   3	{a, b, c}	1
   Total Subsets		8

4. find all subsets of \(D\)
   16 subsets
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{a}, {b}, {c}, {d}	4
   2	{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}	6
   3	{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}	4
   4	{a, b, c, d}	1
   Total Subsets		16

5. fill up the table given below

Set	number of elements	number of subsets	total subsets \(2^{n}\)
\(A\)	1	2	\(2^1 = 2\)
\(B\)	2	4	\(2^2 = 4\)
\(C\)	3	8	\(2^3 = 8\)
\(D\)	4	16	\(2^4 = 16\)

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Let \( N = \{x : x \text{ is counting number up to } 5\} \). Express set \( N \) by the listing method. Make the following subsets from the given set and name them.

\(N = \{1, 2, 3, 4, 5\}\)

1. Subset that has only one element.
   e.g., \(\{1\}\)
2. Subset that has two elements.
   e.g., \(\{1,2\}\)
3. Subset that has three elements.
   e.g., \(\{1,2,3\}\)
4. Subset that has four elements.
   e.g., \(\{1,2,3,4\}\)
5. Subset that has five elements.
   \(\{1,2,3,4,5\}\)
6. Subset having no elements (empty set).
   \(\phi\)
7. Write the number of subsets formed from the given set \( N \).
   \(2^5 = 32\)

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How Many Subsets? Quiz

Given the set below, how many total subsets does it have?

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Power Set

कुनै एक समुह \( S \) को सबै सम्भावित उपसमुहहरूको समुहलाई \( S \) को Power Set भनिन्छ। यसलाई \( P(S) \) द्वारा जनाइन्छ। जस्तै, यदि \( S = \{a, b, c\} \) भने
\( P(S) = \{\phi, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}, \{a, b, c\}\} \)।
जसमा

1. \( n(P(S)) = 2^{n(S)} \)
2. \( S \in P(S) \)

1. Study the given Venn diagram and answer the following questions.
   दिइएको भेन चित्र अध्ययन गर्नुहोस् र निम्न प्रश्नहरूको उत्तर दिनुहोस्।
1. समूहहरु \(R\) र \(S\) को सदस्यहरुलाई सूचिकरण गर्नुहोस ।
   (List the elements of sets \(R\) and \(S\).)[1]
2. समूह \(R\) बाट बन्ने अनुपयुक्त उपसमूहरु लेख्नुहोस ।
   (Write the improper subset formed from the set \(R\).)[1]
3. कुन अवस्थामा दिएको समूहहरु \(R\) र \(S\) अलगिएका समुह बन्छन् ?
   (In which condition, the given sets \(R\) and \(S\) become disjoint?)[1]
1. From the Venn diagram, we get that
   \( R = \{a, b, c\} \)
   \( S = \{c, d, e\} \)
2. The improper subset formed from set \( R \) is
   \( \{a, b, c\} \)
3. Sets \( R \) and \( S \) become disjoint if the element \( c \) is removed from either \( R \) or \( S \) (or both).
2. दिएको भेनचित्र अवलोकन गर्नुहोस। Observe the Given Venn Diagram
1. समूह \(A\) र \(B\) को प्रतिच्छेदन समुहमा कुन कुन सदस्यहरु पर्दछन ?
   (What are the elements of the intersection of set \(A\) and set \(B\)?)[1]
2. समूह \(A = \{a, b, c, e\}\) बाट बन्ने कुनै दुईवटा उपयुक्त उपसमूहहरु लेख्नुहोस ।[1]
   (Write any two proper subsets which can be made from set \(A = \{a, b, c, e\}\).)
3. समूह \(A\) र \(B\) खप्टिएका समुह हुन। यिनलाई अलगिएका समुहहरु बनाउन के गर्नुपर्छ। नयाँ बनेका अलगिएका समुहहरु लेख्नुहोस ।
   (Set \(A\) and set \(B\) are overlapping sets. What should be done to make them disjoint sets? Write the newly formed disjoint sets.)[1]
1. From the Venn diagram, we get that
   \( A \cap B = \{a, b\} \)
2. Any two proper subsets of \( A = \{a, b, c, e\} \) are
   \( \{a, b\} \)
   \( \{c, e\} \)
3. To make sets \( A \) and \( B \) disjoint, we must remove the common elements \( a \) and \( b \) from either \( A \) or \( B \) (or both).
   One possible way is:
   New \( A = \{c, e\} \)
   New \( B = \{d, f\} \)
3. Study the given Venn diagram and answer the following questions
   दिइएको भेन चित्र अध्ययन गर्नुहोस् र निम्न प्रश्नहरूको उत्तर दिनुहोस्।
1. अनुपयुक्त उपसमूहलाई परिभाषा गर्नुहोस ।
   (Define improper subset.)[1]
2. समूह \(B\) को कति ओटा उपसमूह बनाउन सकिन्छ ? सुत्र प्रयोग गरी पत्ता लगाउनुहोस ।
   (How many subsets of set \(B\) can be made? Find using the formula.)[1]
3. समूह \(A\) र \(B\) खप्टिएका वा अलगिएका समूहहरु हुन् ? कारण सहित लेख्नुहोस ।
   (What type of sets are \(A\) and \(B\): overlapping or disjoint sets? Write with reason.)[1]
1. An improper subset of a set is the set itself.
2. From the Venn diagram, set \( B = \{1, 3, 5, 7, 9\} \), so it has 5 elements.
   The number of subsets of a set with \( n \) elements is \( 2^n \).
   Number of subsets of \( B = 2^5 = 32 \)
3. Sets \( A \) and \( B \) are overlapping sets.
   Because they have common elements \( \{1, 3\} \), i.e., \( A \cap B \neq \emptyset \).
4. समूह \(M\) र \(N\) का सदस्यहरू छेउको भेन चित्रमा देखाइएको छ।
   The elements of sets \(M\) and \(N\) are shown in the Venn diagram alongside.
1. समूह \(M\) र \(N\) का सबै सम्भावित उचित (proper) र अनुचित (improper) उपसमूहहरू लेख्नुहोस्।
   Write all possible proper and improper subsets of sets \(M\) and \(N\).[1]
2. दिइएका समूह \(M\) र \(N\) का उपसमूहहरूको संख्या बराबर छ वा छैन, लेख्नुहोस?
   Are the subsets of the given sets \(M\) and \(N\) equal in number?[1]
3. समूह \(M\) को कुन सदस्य हटाउँदा समुहहरु \(M\) र \(N\) अलगिएका समुह बन्छन्?
   Which member of set \(M\) must be removed to make \(M\) and \(N\) disjoint sets?[1]
1. From the Venn diagram:
   \( M = \{a, b, c\} \)
   \( N = \{b, d, f\} \)
   
   Subsets of \( M \) are
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{a}, {b}, {c}	3
   2	{a, b}, {a, c}, {b, c}	3
   3	{a, b, c}	1
   Total Subsets		8

   Subsets of \( N \) are
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{b}, {d}, {f}	3
   2	{b, d}, {b, f}, {d, f}	3
   3	{b, d, f}	1
   Total Subsets		8

2. Yes, the number of subsets of sets \( M \) and \( N \) are equal.
   Both sets have 3 elements, so each has \( 2^3 = 8 \) subsets.
3. To make \( M \) and \( N \) disjoint, the common element \( b \) must be removed from set \( M \).
   Removing \( b \) from \( M \) gives \( M = \{a, c\} \), and then \( M \cap N = \emptyset \).
5. प्रश्नमा (Given)
   \(P = \{\text{Prime numbers less than } 10\}\)
   \(Q = \{\text{Odd numbers less than } 8\}\)
   \(R = \{\text{Prime factors of } 6\}\)
   \(S = \{\text{Cube numbers between } 10 \text{ and } 20\}\)
1. माथिका चार समूहहरूमध्ये कुन-कुन समुहहरु समतुल्य (equivalent) समूह हुन्?
   Which of the four sets are equivalent sets?[1]
2. समूह \(R\) का उपसमूहहरू लेख्नुहोस्।
   Write the subsets of set \(R\).[1]
3. समूह \(P\) को उपयुख्त उपसमूह समूह \(S\) हो वा होईन? कारणसहित लेख्नुहोस्।
   Is the set \(S\) a proper subset of set \(P\)? Write with reason.[1]
1. First, list all sets:
   \( P = \{2, 3, 5, 7\} \) → 4 elements
   \( Q = \{1, 3, 5, 7\} \) → 4 elements
   \( R = \{2, 3\} \) → 2 elements
   \( S = \{\} = \emptyset \) → 0 elements (no cube numbers between 10 and 20)
   
   Equivalent sets have the same number of elements.
   So, \( P \) and \( Q \) are equivalent sets.
2. Set \( R = \{2, 3\} \)
   Subsets of \( R \) are:
   \( \emptyset, \{2\}, \{3\}, \{2, 3\} \)
3. Yes, \( S \) is a proper subset of \( P \), because \( S = \emptyset \).
6. एक भेन चित्र दिइएको छ। Given a Venn diagram
1. उपयूक्त उपसमूह भन्नाले के जनाउँछ?
   Define proper subset.[1]
2. समुह \( L \) को अनुपयुक्त उपसमूह लेख्नुहोस ।
   Write the improper subset of \( L \).[1]
3. यदि सदस्यहरु \( a, e , i ,o , u \) समुह \( M \) को मात्र हुन् भने, \( L \) र \( M \) कस्ता प्रकारका समूह हुन्? कारणसहित लेख्नुहोस्।
   If \( a, e , i ,o , u \) are the members of set \( M \) only, then what type of set are \( L \) and \( M \)? Write with reason.[1]
1. A proper subset of a set is a subset that contains some, but not all, elements of the given set. In other words,
   Set \( A \) is a proper subset of set \( B \) if every element of \( A \) is in \( B \), and \( A \neq B \)
2. From the Venn diagram, set \( L = \{a, b, c\} \).
   The improper subset of \( L \) is the set L itself, which is
   \( \{a, b, c\} \)
3. If \( a, e, i, o, u \) are members of set \( M \) only, then, \( L \) and \( M \) are disjoint sets because \( L \cap M = \emptyset \).
7. संगैको भेन चित्रलाई हेरि तलका प्रश्नहरूको उत्तर दिनुहोस्।
   Look at the adjoining Venn diagram and answer the following questions:
1. समुह \( A \) र \( B \) अलगिएका समुह हुन की खप्टिएका पहिचान गर्नुहोस्।
   Identify whether the sets \( A \) and \( B \) are disjoint or overlapping.[1]
2. समुह \( B \) का उपयुक्त उप-समुहहरूको सूची तयार गर्नुहोस्।
   Prepare the list of the proper subsets of set \( B \).[1]
3. दिइएका समुहहरूलाई अलगिएका समुह बनाउन के कस्तो समायोजन आवश्यक छ?
   What adjustments are needed to ensure the given sets are disjoint?[1]
1. From the Venn diagram, sets \( A \) and \( B \) share common elements \( \{4, 5\} \).
   Therefore, \( A \) and \( B \) are overlapping sets.
2. From the diagram, set \( B = \{4, 5, 6, 7\} \).
   Proper subsets of \( B \) are all subsets except \( B \) itself. So, list of the proper subsets of set \( B \) are
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{4}, {5}, {6}, {7}	4
   2	{4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}	6
   3	{4, 5, 6}, {4, 5, 7}, {4, 6, 7}, {5, 6, 7}	4
   Total		15

3. To make sets \( A \) and \( B \) disjoint, the common elements \( 4 \) and \( 5 \) must be removed from either \( A \) or \( B \) (or both).
8. संगैको भेन चित्रलाई हेरि तलका प्रश्नहरूको उत्तर दिनुहोस्।
   Look at the adjoining Venn diagram and answer the following questions:
1. समुह \( A \) र \( B \) अलगिएका समुह हुन की खप्टिएका पहिचान गर्नुहोस्।
   Identify whether the sets \( A \) and \( B \) are disjoint or overlapping.[1]
2. समुह \( B \) का उपयुक्त उप-समुहहरूको सूची तयार गर्नुहोस्।
   Prepare the list of the proper subsets of set \( A \).[1]
3. दिइएका समुहहरूलाई खप्टिएका समुह बनाउन के कस्तो समायोजन आवश्यक छ?
   What adjustments are needed to ensure the given sets are overlaping?[1]
1. From the Venn diagram, the circles for sets \( A \) and \( B \) do not overlap, and there are no common elements shown between them.
   Therefore, \( A \) and \( B \) are disjoint sets.
2. From the diagram, set \( A = \{x, y, z\} \).
   Proper subsets of \( A \) are all subsets except \( A \) itself. which are
   

   Number of Elements	Subsets	Count
   0	∅	1
   1	{x}, {y}, {z}	3
   2	{x, y}, {x, z}, {y, z}	3
   Total Subsets		7

3. It is needed to have some ellement in common then
   Thus, \( A \cap B \ne \emptyset \)
9. संगैको भेन चित्रलाई हेरि तलका प्रश्नहरूको उत्तर दिनुहोस्।
   Study the Venn diagram given alongside and answer the following questions:
1. समुह \( P \) र \( Q \) अलगिएका छन् कि खप्टिएका छन पहिचान गर्नुहोस्।
   Identify whether the sets \( P \) and \( Q \) are disjoint or overlapping.[1]
2. समुह \( P \) र \( Q \) का साझा सदस्यहरुको समुह के हो?
   What is the set of common elements in sets \( P \) and \( Q \)?[1]
3. समुह \( P \) बाट कुन सदस्यहरु हटाउदा समुह \( P \) र \( Q \) अलगिएका समुह हुन्छन?
   Which elements from the set \( P \) need to be removed in order to make sets \( P \) and \( Q \) disjoint?[1]
1. From the Venn diagram, sets \( P \) and \( Q \) share the elements \( 6 \) and \( 7 \) in the overlapping region.
   Therefore, \( P \) and \( Q \) are overlapping sets.
2. The common elements of sets \( P \) and \( Q \) are the elements in the intersection region.
   \( P \cap Q = \{6, 7\} \)
3. To make sets \( P \) and \( Q \) disjoint, the common elements must be removed from set \( P \).
   Remove \( 6 \) and \( 7 \) from \( P \).
10. Study the given Venn diagram and answer the following questions.
1. List the elements of \( A \).(1)
2. If there were no common members in \( A \) and \( B \), what types of sets would they be?(1)
3. Write the improper subset of set \( B \).(1)
1. From the Venn diagram, the elements of set \( A \) are:
   \( A = \{1, 2, 3, 5, 7\} \)
2. If there were no common members in \( A \) and \( B \), then they would be disjoint sets.
3. From the diagram, set \( B = \{1, 5, 8, 9, 10\} \).
   The improper subset of \( B \) is the set itself, which is
   \( \{1, 5, 8, 9, 10\} \)
11. दिएको भेन चित्र अध्ययन गर्नुहोस् र निम्नलिखित प्रश्नहरूको उत्तर दिनुहोस ।
    Study the given Venn diagram and answer the following questions.
1. समूह A का सदस्यहरू लेख्नुहोस |
   List the elements of set A.(1)
2. समूह A र B को प्रतिच्छेदन \((A \cap B)\) का सदस्यहरू लेख्नुहोस|
   List the elements of the intersection of sets A and B i.e. \((A \cap B)\).(1)
3. समूह A मा रहेका तर B मा नरहेका सदस्यहरू \((A-B)\) लेख्नुहोस|
   List the elements that are in set A but not in set B i.e. \((A-B)\).
1. From the Venn diagram, the elements of set \( A \) are:
   \( A = \{2, 3, 5, 7, 9\} \)
2. The intersection \( A \cap B \) consists of elements common to both \( A \) and \( B \).
   \( A \cap B = \{2\} \)
3. The elements in \( A \) but not in \( B \) are those only in the left part of circle \( A \):
   \( A - B = \{3, 5, 7, 9\} \)
12. दिएको भेनचित्र हेरी तल्ला प्रश्नहरूको उत्तर दिनुहोस|
    Look at the given Venn diagram and answer the following questions.
1. समूह \(A\) र समूह \(B\) कस्ता प्रकारको समुहहरू हुन?
   What types of sets are Set \(A\) and Set \(B\)?(1)
2. समूह \(B\) बाट कति ओटा उपसमुहहरू बन्छन?
   How many subsets can be formed from Set \(B\)?(1)
3. \(A\) र \(B\) सर्वव्यापक समूह \(U\) का उपसमूहहरू भए यसलाई सांकेतिक रूपमा लेख्नुहोस |(1)
   If \(A\) and \(B\) are subsets of the universal set \(U\), write this in symbolic form.
1. From the Venn diagram, sets \( A \) and \( B \) have common elements \( \{e, f\} \).
   Therefore, \( A \) and \( B \) are overlapping sets.
2. From the diagram, set \( B = \{b, e, f, g, h\} \), which has 5 elements.
   The number of subsets of a set with \( n \) elements is \( 2^n \).
   Number of subsets of \( B = 2^5 = 32 \)
3. Since both \( A \) and \( B \) are contained within the universal set \( U \), we write:
   \( A \subseteq U \) and \( B \subseteq U \)
13. दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
    From the given Venn diagram answer the following questions.
1. उपयुक्त उपसमूह भनेको के हो ?
   What is proper subset?(1)
2. \( P \) का सदस्य सङ्ख्या लेख्नुहोस् |
   Write the number of elements of \( P \).(1)
3. \( Q \) बाट बन्ने सबै उपसमूहहरु बनाउनुहोस् |
   Make all subsets of set \( Q \).(1)
1. A proper subset of a set is a subset that contains some but not all elements of the original set. In other words, set \( A \) is a proper subset of set \( B \) if every element of \( A \) is in \( B \), and \( A \neq B \).
   Symbolically, \( A \subset B \) and \( A \neq B \).
2. From the Venn diagram, set \( P = \{a, b, c,d, e\} \).
   Number of elements in \( P = 5 \)
3. From the diagram, set \( Q = \{c, d\} \).
   All subsets of \( Q \) are:
   \( \emptyset, \{c\}, \{d\}, \{c, d\} \)
14. दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
    From the given Venn diagram answer the following questions.
1. समुह A को सदस्यहरु लेख्नुहोस ?
   Write the elements of set A(1)
2. \( A \) र \( B \) को सम्बन्ध संकेतमा लेख्नुहोस् |
   Write relation between A and B in symbol.(1)
3. \( B \) मा नभएका सदस्यहरु लेख्नुहोस |
   Write elements which are not in B(1)
1. From the Venn diagram, the elements of set \( A \) are:
   \( A = \{1, 2, 3,4,5, 6\} \)
2. Sets \( B \) is subset of \( A \)
   \( B \subset A \)
3. The elements not in set \( B \) are
   \( \{1,2,5,6\} \)
15. दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
    From the given Venn diagram answer the following questions.
1. समुह A को सदस्य सँख्या लेख्नुहोस ?
   Write the cardinality of set A(1)
2. उपसमुह भनेको के हो? लेख्नुहोस् |
   What is subset, write.(1)
3. समुह \( A \) बाट बन्ने 2 सदस्य हुने सबै उपसमुहहरु लेख्नुहोस |
   Write all subsets of A containing 2 elements(1)
1. From the Venn diagram, set \( A = \{q,r,s, t\} \).
   Cardinality of set \( A = 4 \)
2. A subset is a set whose every element is also an element of another set. If all elements of set \( A \) are in set \( B \), then \( A \) is a subset of \( B \), written as \( A \subseteq B \)
3. Set \( A = \{q,r,s, t\} \) has 4 elements, so any twwo subsets of A are
   \(\{s, t\} \) and \(\{q, r\} \)
16. दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
    From the given Venn diagram answer the following questions.
1. भेनचित्रमा कुन प्रकारका समूहहरू छन् ?
   Which type of sets are shown in the Venn diagram?(1)
2. \( A \) का सदस्य सङ्ख्या लेख्नुहोस् |
   Write the number of elements of \( A \).(1)
3. \( B \) बाट बन्ने कुनै एक उपसमूह लेख्नुहोस् |
   Make one subset of set \( B \).(1)
1. From the Venn diagram, sets \( A \) and \( B \) have no common elements (no overlapping region with elements).
   Therefore, \( A \) and \( B \) are disjoint sets.
2. From the Venn diagram, set \( A = \{1, 2, 3\} \).
   Number of elements of \( A = 3 \)
3. From the diagram, set \( B = \{5, 7, 9\} \).
   One subset of \( B \) is:
   \( \{5, 7\} \)
17. समूहहरू \(P = \{l, o, v, e\}\) र \(Q = \{h, a, t, e\}\) दिइएको छ । (Given the set \(P = \{l, o, v, e\}\) and \(Q = \{h, a, t, e\}\).)
1. समूह \(P\) र \(Q\) खप्टिएको वा अलगिएका कस्ता खालका समूह हुन् ? लेख्नुहोस् ।
   (What types of set \(P\) and set \(Q\) are—overlapping or disjoint? Write.)[1]
2. \(Q\) को 2 वटा उपयुक्त उपसमूहहरू लेख्नुहोस् ।
   (Write any two proper subsets of set \(Q\).)[1]
3. \(P\) र \(Q\) लाई अलगिएका समूह बनाउन समूह \(P\) बाट कुन सदस्य हटाउनु पर्छ ?
   (What member of set \(P\) need to be removed to make set \(P\) and \(Q\) disjoint sets?)[1]
1. Sets \( P = \{l, o, v, e\} \) and \( Q = \{h, a, t, e\} \) both contain the common element \( e \).
   Therefore, \( P \) and \( Q \) are overlapping sets.
2. Two proper subsets of \( Q = \{h, a, t, e\} \) are:
   \( \{h, a\} \)
   \( \{t\} \)
3. To make \( P \) and \( Q \) disjoint, the common element \( e \) must be removed from set \( P \).
   Remove \( e \) from \( P \).
18. Answer the following questions on the basis of the Venn diagram alongside.
1. Write whether sets \(N\) and \(M\) are overlapping or disjoint?[1]
2. Write an improper subset of set \(N\).[1]
3. Which social media to be removed such that sets \(N\) and \(M\) are disjoint?[1]
1. From the Venn diagram, sets \( N \) and \( M \) share the common element "TT" in the overlapping region.
   Therefore, \( N \) and \( M \) are overlapping sets.
2. From the diagram, set \( N = \{F, TT\} \).
   The improper subset of \( N \) is the set itself:
   \( \{F, TT\} \)
3. To make sets \( N \) and \( M \) disjoint, the common element "TT" must be removed from set \( N \) (or from \( M \)).
   Remove "TT" from set \( N \).
19. सर्वव्यापक समूह \( U = \{a, b, c, d, e\} \) का दुई उपसमूहहरू \( D = \{a, b, c\} \) र \( E = \{a, b, c, d, e\} \) छन् ।
    (Given universal set \( U = \{a, b, c, d, e\} \), two subsets are \( D = \{a, b, c\} \) and \( E = \{a, b, c, d, e\} \).)
1. D र E मध्ये कुन उपयुक्त र कुन अनुपयुक्त उपसमूह हुन्, लेख्नुहोस् ।
   (Between D and E, which one is the proper subset and which one is the improper subset of U? Write it.)[1K]
2. समूह E का कति ओटा उपसमूह बन्छन् ? लेख्नुहोस् ।
   (How many subsets of set E can be formed? Write it.)[1HA]
3. समूहहरू D र E लाई भेनचित्रमा देखाउनुहोस् र यी दुई समूह कस्ता समूह हुन्, लेख्नुहोस् ।
   (Show sets D and E in a Venn diagram and write what type of sets are D and E.)[1U]
1. From the given sets:
   \( D = \{a, b, c\} \) is a proper subset of \( U \) because \( D \subset U \) and \( D \neq U \).
   \( E = \{a, b, c, d, e\} \) is an improper subset of \( U \) because \( E = U \).
2. Set \( E \) has 5 elements.
   Number of subsets of \( E = 2^5 = 32 \)
3. Sets \( D \) and \( E \) in Venn-diagram
   
   Here
   Since all elements of \( D \) are in \( E \), the set \( D \) is a subset of \( E \)
   
20. A र B सर्वव्यापक समूह \( U =\) {x : x एउटा 20 भन्दा सानो प्राकृतिक सङ्ख्या हुन्} का उपसमूहहरू हुन् । यदि \( A \)= 2 का अपवर्त्यहरू} र \( B = \) रूढ सङ्ख्याहरू} छन् ।
    (A and B are subsets of the universal set \( U = \{x : x \text{ is a natural number less than } 20\} \). If \( A = \{\text{multiples of } 2\} \) and \( B = \{\text{prime numbers}\} \).)
1. समूह A र B लाई सूचीकरण विधिमा लेख्नुहोस् र यी खण्डिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
   (List the elements of sets A and B, then write whether these two sets are overlapping sets or disjoint sets.)[1A]
2. माथिका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
   (Show the above sets in a Venn diagram.)[1U]
3. समूह A का कुनै चारओटा चार सदस्यिय उपसमूहहरू बनाउनुहोस् ।
   (Make four subsets of set A having four members.)[1HA]
1. Elements of sets:
   \( A = \{2, 4, 6, 8, 10, 12, 14, 16, 18\} \)
   \( B = \{2, 3, 5, 7, 11, 13, 17, 19\} \)
   They share element 2 → so they are overlapping sets.
2. Venn diagram:
   
3. Four subsets of A with four members are
   \( \{2, 4, 6, 8\} \)
   \( \{4, 6, 8, 10\} \)
   \( \{8, 10, 12, 14\} \)
   \( \{10, 12, 14, 16\} \)
21. सर्वव्यापक समूह \( U = \{1, 2, 3, 4, 5, 6\} \) का उपसमूहहरू \( P = \{2, 3, 4\} \) र \( Q = \{1, 2, 4\} \) छन् ।
    (P = {2, 3, 4} and Q = {1, 2, 4} are subsets of the universal set U = {1, 2, 3, 4, 5, 6}.)
1. समूहहरू P र Q खण्डिएका वा अलगिएका कस्ता समूह हुन्? लेख्नुहोस् ।
   (What type of sets are P and Q — overlapping or disjoint?)[1K]
2. समूह P का दुई सदस्यिय सबै उपसमूहहरू लेख्नुहोस् ।
   (Write all subsets of set P with two members.)[1A]
3. समूहहरू U, P र Q लाई भेनचित्रमा देखाउनुहोस् ।
   (Show sets U, P and Q in a Venn diagram.)[1U]
1. Set \( P = \{2, 3, 4\} \), Set \( Q = \{1, 2, 4\} \)
   Common elements: 2 and 4 → so they are overlapping sets.
2. All 2-member subsets of P are
   \( \{2, 3\} \)
   \( \{2, 4\} \)
   \( \{3, 4\} \)
3. Venn diagram is as below
   
22. दिइएको भेनचित्र अध्ययन गरी तल सोधिएका प्रश्नहरूको उत्तर दिनुहोस् ।
    (Study the given Venn diagram and answer the questions asked below.)
1. सर्वव्यापक समूह U लाई सूचीकरण विधिबाट लेख्नुहोस् ।
   (List the elements of the universal set U.)[1A]
2. M र N अलगिएका वा खण्डिएका कस्ता समूह हुन्, कारणसहित लेख्नुहोस् ।
   (What type of sets are M and N — overlapping or disjoint sets? Write with reason.)[1A]
3. समूह M का कति ओटा उपसमूह बनाउन सकिन्छ? लेख्नुहोस् ।
   (How many subsets of set M can be formed? Write it.)[1HA]
1. From the Venn diagram, the universal set \( U \) contains all elements shown inside and outside the circles.
   \( U = \{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\} \)
2. Sets \( M \) and \( N \) share common elements \( 6 \) and \( 12 \) in the overlapping region.
   Therefore, \( M \) and \( N \) are overlapping sets.
3. Set \( M = \{3, 6, 9, 12, 15\} \), which has 5 elements.
   Number of subsets of \( M = 2^5 = 32 \)
23. सर्वव्यापक समूह \( U = \{1, 2, 3, 4, 5, 6\} \) का दुईओटा उपसमूहहरू \( A = \{1, 3, 4, 5\} \) र \( B = \{2, 3, 5\} \) छन् ।
    (Two subsets of the universal set \( U = \{1, 2, 3, 4, 5, 6\} \) are \( A = \{1, 3, 4, 5\} \) and \( B = \{2, 3, 5\} \).)
1. समूह A र B खण्डिएका वा अलगिएका कस्ता समूह हुन्? कारणसहित लेख्नुहोस् ।
   (Are sets A and B overlapping or disjoint sets? Write with reason.)[1K]
2. समूह A का एक सदस्यिय उपसमूहहरू लेख्नुहोस् ।
   (Write all the subsets of set A having single element.)[1A]
3. समूहहरू U, A र B लाई भेनचित्रमा देखाउनुहोस् ।
   (Show sets U, A and B in a Venn diagram.)[1U]
1. Sets \( A = \{1, 3, 4, 5\} \) and \( B = \{2, 3, 5\} \) have common elements \( 3 \) and \( 5 \).
   Since \( A \cap B = \{3, 5\} \neq \emptyset \), they are overlapping sets.
2. All single-element subsets of set \( A = \{1, 3, 4, 5\} \) are:
   \( \{1\}, \{3\}, \{4\}, \{5\} \)
3. Venn diagram is as below
   
24. सर्वव्यापक समूह \( U = \{a, e, i, o, u\} \) का दुईओटा उपसमूहहरू \( R = \{a, i, o\} \) र \( S = \{i, o, u\} \) छन् ।
    (Two subsets of the universal set \( U = \{a, e, i, o, u\} \) are \( R = \{a, i, o\} \) and \( S = \{i, o, u\} \).)
1. समूह R र S खण्डिएका वा अलगिएका कस्ता समूह हुन्? लेख्नुहोस् ।
   (What type of sets are R and S — overlapping or disjoint?)[1K]
2. माथिका समुहहरुलाई भेनचित्रमा देखाउनुहोस् ।
   (Show above sets in venn-diagram)[1A]
3. समूह R र S का कुनै दुईवटा साझा उपसमुहहरु लेख्नुहोस् ।
   (Wtite any two subsets which are formed from both R and S)[1U]
1. Sets \( R = \{a, i, o\} \) and \( S = \{i, o, u\} \) share common elements \( i \) and \( o \).
   Since \( R \cap S = \{i, o\} \neq \emptyset \), they are overlapping sets.
2. The venn-diagram is as below.
3. The common elements of \( R \) and \( S \) are \( \{i, o\} \). Any subsets formed from these common elements are subsets of both \( R \) and \( S \). Two such subsets are:
   \( \{i\} \)
   \( \{i, o\} \)
25. सर्वव्यापक समूह \( U =\) {x : x एउटा 8 भन्दा सानो प्राकृतिक सङ्ख्या हो} का उपसमूहहरू \( F = \{1, 2, 3\} \) र \( G = \{2, 3, 5\} \) छन् ।
    (The subsets of the universal set \( U = \{x : x \text{ is a natural number less than } 8\} \) are \( F = \{1, 2, 3\} \) and \( G = \{2, 3, 5\} \).)
1. सर्वव्यापक समूह U लाई सूचीकरण विधिबाट लेख्नुहोस् र समूहहरू F र G कस्ता समूह हुन्, कारणसहित लेख्नुहोस् ।
   (Write the universal set U in a listing method. Also write with reason that what type of sets F and G are — disjoint or overlapping sets.)[1A]
2. समूहहरू F र G का कति ओटा उपसमूह बनाउन सकिन्छ? लेख्नुहोस् ।
   (How many subsets of set F and G can be made?)[1HA]
3. कुनै समूहको अनुपयुक्त उपसमूह भनेको कस्तो समूह हो ?
   (What is the improper subset of a set?)[1K]
1. Natural numbers less than 8 are: 1, 2, 3, 4, 5, 6, 7.
   So, \( U = \{1, 2, 3, 4, 5, 6, 7\} \)
   Sets \( F = \{1, 2, 3\} \) and \( G = \{2, 3, 5\} \) share common elements \( 2 \) and \( 3 \).
   Since \( F \cap G = \{2, 3\} \neq \emptyset \), they are overlapping sets.
2. Set \( F \) has 3 elements → Number of subsets = \( 2^3 = 8 \)
   Set \( G \) has 3 elements → Number of subsets = \( 2^3 = 8 \)
   So, 8 subsets can be formed from each of F and G.
3. An improper subset of a set is the set itself.
   For example, the improper subset of \( \{1, 2, 3\} \) is \( \{1, 2, 3\} \).
26. A र B सर्वव्यापक समूह U का उपसमूहहरू हुन् । यदि \( U =\) {x : x एउटा 10 भन्दा सानो पूर्ण सङ्ख्या हो} , \( A =\) {y : y एउटा 10 भन्दा सानो विषम सङ्ख्या हो} र \( B =\) {z : z एउटा 10 भन्दा सानो रूढ सङ्ख्या हो} ।
    (A and B are the subsets of the universal set U. If \( U = \{x : x \text{ is a whole number less than } 10\} \), \( A = \{y : y \text{ is an odd number less than } 10\} \) and \( B = \{z : z \text{ is a prime number less than } 10\} \).)
1. माथिका समूहहरूलाई सूचीकरण विधिमा लेख्नुहोस् ।
   (List the elements of the above sets.)[1A]
2. समुह A र B खप्टिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
   (What type of sets are A and B — overlapping or disjoint? Write it.)[1K]
3. समूह B का दुई सदस्यिय उपसमूहहरू लेख्नुहोस् ।
   (Write the subsets of set B with two elements.)[1HA]
1. Whole numbers less than 10: \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \)
   Odd numbers less than 10: \( 1, 3, 5, 7, 9 \)
   Prime numbers less than 10: \( 2, 3, 5, 7 \)
   \( U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
   \( A = \{1, 3, 5, 7, 9\} \)
   \( B = \{2, 3, 5, 7\} \)
2. Sets \( A \) and \( B \) share common elements \( 3, 5, 7 \).
   Since \( A \cap B = \{3, 5, 7\} \neq \emptyset \), they are overlapping sets.
3. Set \( B = \{2, 3, 5, 7\} \).
   Some 2-element subsets of \( B \) are:
   \( \{2, 3\} \)
   \( \{5, 7\} \)
27. एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)
1. अनुपयुक्त उपसमूह लाई परिभाषित गर्नुहोस् । Define improper subset.[1K]
2. समूह B का 2 सदस्यिय उपयुक्त उपसमूहहरू लेख्नुहोस् ।
   (Write proper subsets of set B with two members.)[1U]
3. समूहहरू A र B खण्डिएका समूहहरू हुन् । समर्थन गर्नुहोस् ।
   (Sets A and B are overlapping sets. Justify it.)[1HA]
1. अनुपयुक्त उपसमूह लाई परिभाषित गर्नुहोस् । (Define improper subset.)[1K]
   An improper subset is a subset of a set that includes all the elements of the original set, i.e., the set itself.
2. समूह B का 2 सदस्यिय उपयुक्त उपसमूहहरू लेख्नुहोस् ।
   (Write proper subsets of set B with two members.)[1U]
   The proper subsets of set S with two members are {9, 10}, {9, 12}, and {10, 12}.
3. समूहहरू A र B खण्डिएका समूहहरू हुन् । समर्थन गर्नुहोस् ।
   (Sets A and B are overlapping sets. Justify it.)[1HA]
   Sets A and B are overlapping because they have a common element, 9, as indicated by their non-empty intersection (R ∩ S = {9}).
28. P = {a, r, e} र Q = {a, r, t} सर्वव्यापक समूह U = {a, r, t, e, s} का उपसमूहहरू हुन् ।
    (P = {a, r, e} and Q = {a, r, t} are subsets of the universal set U = {a, r, t, e, s}.)
1. उपयुक्त समूहलाई परिभाषित गर्नुहोस् । Define proper subset.[1K]
2. यो कुन समूह हो जुन सबै समूहहरूको उपसमूह हो ?
   (What is the set which is subset of every set?)[1U]
3. यदि समूह P = {e, s} भएको भए, समूहहरू P र Q बीचको सम्बन्ध कस्तो हुन्थ्यो ? कारणसहित लेख्नुहोस् ।
   (If set P = {e, s}, what will be the relation between set P and Q? Write with reason.)[1HA]
1. उपयुक्त समूहलाई परिभाषित गर्नुहोस् । (Define proper subset.)[1K]
   A proper subset of a set is a subset that contains some, but not all, elements of the original set. In other words, if \( A \) is a proper subset of \( B \), then \( A \subseteq B \) and \( A \neq B \).
   For example, \( \{a, r\} \) is a proper subset of \( \{a, r, e\} \).
2. यो कुन समूह हो जुन सबै समूहहरूको उपसमूह हो ?
   (What is the set which is subset of every set?)[1U]
   The empty set, denoted \( \emptyset \), is a subset of every set.
3. यदि समूह P = {e, s} भएको भए, समूहहरू P र Q बीचको सम्बन्ध कस्तो हुन्थ्यो ? कारणसहित लेख्नुहोस् ।
   (If set P = {e, s}, what will be the relation between set P and Q? Write with reason.)[1HA]
   If \( P = \{e, s\} \) and \( Q = \{a, r, t\} \), then (\( P \cap Q = \emptyset \)), so the sets \( P \) and \( Q \) are disjoint sets.
   
29. सर्वव्यापक समूह \( U = \{1, 2, 3, 4, 5\} \) का दुईओटा उपसमूहहरू \( D = \{1, 2, 3\} \) र \( E = \{3, 4, 5\} \) छन् ।
    (Two subsets of the universal set \( U = \{1, 2, 3, 4, 5\} \) are \( D = \{1, 2, 3\} \) and \( E = \{3, 4, 5\} \).)
1. खप्टिएका समूहलाई परिभाषित गर्नुहोस् । Define overlapping sets.[1K]
2. D र E खप्टिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
   (Write common subsets of the sets D and E.)[1U]
3. समूह D र E बाट कुन सदस्य हटाउँदा यी समूहहरू अलगिएका समूह बन्छन् ?
   (Which element of set D and set E should be removed to make them disjoint sets?)[1HA]
1. खप्टिएका समूहलाई परिभाषित गर्नुहोस् । (Define overlapping sets.)[1K]
   Overlapping sets are sets that have at least one common element. In other words, if two sets \( A \) and \( B \) are overlapping, then their intersection is not empty, i.e., \( A \cap B \neq \emptyset \).
   
2. D र E खप्टिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
   (Write common subsets of the sets D and E.)[1U]
   The common elements of \( D = \{1, 2, 3\} \) and \( E = \{3, 4, 5\} \) are \( D \cap E = \{3\} \). So,
   D and E are overlaping sets.
3. समूह D र E बाट कुन सदस्य हटाउँदा यी समूहहरू अलगिएका समूह बन्छन् ?
   (Which element of set D and set E should be removed to make them disjoint sets?)[1HA]
   Sets \( D = \{1, 2, 3\} \) and \( E = \{3, 4, 5\} \) have a common element, \( 3 \), since \( D \cap E = \{3\} \). So, removing the element \( 3 \) from either \( D \) or \( E \) makes them disjoint sets.
30. सँगै दिइएको भेनचित्र अध्ययन गरी तलका प्रश्नहरूको उत्तर दिनुहोस् ।
    (Study the given Venn diagram, then answer the following questions.)
1. कस्ता समूहहरूलाई अलगिएका समूह भनिन्छ ? लेख्नुहोस् ।
   (What type of sets are disjoint sets? Write.)[1K]
2. समूह R का सबै एक सदस्यिय उपसमूहहरू लेख्नुहोस् ।
   (Write all subsets of set R with single elements.)[1U]
3. कुन अवस्थामा दिइएका समूहहरू R र S अलगिएका समूह बन्छन् ?
   (In what condition, the given sets R and S become disjoint sets?)[1HA]
1. कस्ता समूहहरूलाई अलगिएका समूह भनिन्छ ? (What type of sets are disjoint sets? Write.)[1K]
   Disjoint sets are sets that have no common elements. In other words, if two sets \( A \) and \( B \) are disjoint, then their intersection is empty, i.e., \( A \cap B = \emptyset \).
2. समूह R का सबै एक सदस्यिय उपसमूहहरू लेख्नुहोस् ।
   (Write all subsets of set R with single elements.)[1U]
   From the Venn diagram, set \( R = \{a, b\} \). The single-element subsets of \( R \) are:
   \( \{a\}, \{b\} \).
3. कुन अवस्थामा दिइएका समूहहरू R र S अलगिएका समूह बन्छन् ?
   (In what condition, the given sets R and S become disjoint sets?)[1HA]
   From the Venn diagram, \( R = \{a, b\} \) and \( S = \{a,b,c, d\} \), with \( R \cap S = \{a, b\} \). To make \( R \) and \( S \) disjoint, R must not contain any elements from \( S = \{a,b,c, d\} \), it can contain elements from \( \{e, f\}\).
31. A = {x : x ≤ 3, x ∈ N} र B = {y : y ≤ 4, y ∈ N} सर्वव्यापक समूह U = {z : z ≤ 6, z ∈ W} का उपसमूहहरू हुन् ।
    (A = {x : x ≤ 3, x ∈ N} and B = {y : y ≤ 4, y ∈ N} are subsets of the universal set U = {z : z ≤ 6, z ∈ W}.)
1. समूहहरू A, B र U लाई सूचीकरण विधिबाट लेख्नुहोस् ।
   (Write the sets A, B and U in listing method.)[1A]
2. समूहहरू A र B का दुई सदस्यिय साझा उपसमूहहरू लेख्नुहोस् ।
   (Write all the common subsets of set A and B with two elements.)[1U]
3. समूह B को कुन सदस्य हटाउँदा A र B एकअर्काको अनुपयुक्त उपसमूह बन्छन् ?
   (Which element of set B is to be removed so that they become improper subset of A and B each other?)[1HA]
1. Natural numbers (N) start from 1, and whole numbers (W) include 0.
   \( A = \{x : x \leq 3, x \in \mathbb{N}\} = \{1, 2, 3\} \)
   \( B = \{y : y \leq 4, y \in \mathbb{N}\} = \{1, 2, 3, 4\} \)
   \( U = \{z : z \leq 6, z \in \mathbb{W}\} = \{0, 1, 2, 3, 4, 5, 6\} \)
2. Common elements of \( A \) and \( B \): \( A \cap B = \{1, 2, 3\} \)
   All 2-element subsets of this intersection are the common 2-element subsets of both \( A \) and \( B \), which are
   \( \{1, 2\} \)
   \( \{1, 3\} \)
   \( \{2, 3\} \)
3. Currently, \( A = \{1, 2, 3\} \) and \( B = \{1, 2, 3, 4\} \).
   For \( A \) and \( B \) to be improper subsets of each other, the element \( 4 \) must be removed from set \( B \)
32. एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)
1. कस्तो समूहलाई दिइएको समूहको उपसमूह भनिन्छ ? लेख्नुहोस् ।
   (What type of set is called subset of the given set? Write it.)[1K]
2. समूहहरू A, B र U बीच कस्तो सम्बन्ध छ ? लेख्नुहोस् ।
   (What is the relation between sets A, B and U? Write it.)[1U]
3. समूहहरू A र B लाई बराबर समूह बनाउन, समूह A बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
   (To make sets A and B equal, which elements of set A are to be removed?)[1HA]
1. कस्तो समूहलाई दिइएको समूहको उपसमूह भनिन्छ ? (What type of set is called subset of the given set? Write it.)[1K]
   A set is called a subset of a given set if all its elements are also elements of the given set. In other words, if every element of set \( A \) is contained in set \( B \), then \( A \subseteq B \)
2. समूहहरू A, B र U बीच कस्तो सम्बन्ध छ ? लेख्नुहोस् ।
   (What is the relation between sets A, B and U? Write it.)[1U]
   From the Venn diagram, \( U = \{a, b, c, d, e, f\} \), \( A = \{a, b, c, d\} \), and \( B = \{a, b\} \). Since all elements of \( A \) and \( B \) are in \( U \), and all elements of \( B \) are in \( A \), the relations are follows.
   \( B \subseteq A \) and \( A \subseteq U \), \( B \subseteq U \).
3. समूहहरू A र B लाई बराबर समूह बनाउन, समूह A बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
   (To make sets A and B equal, which elements of set A are to be removed?)[1HA]
   From the Venn diagram, \( A = \{a, b, c, d\} \) and \( B = \{a, b\} \). To make \( A \) equal to \( B \), the elements \( c \) and \( d \) must be removed from \( A \)
33. एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)
1. P र Q खण्डिएका वा अलगिएका कस्ता समूह हुन् ? कारणसहित लेख्नुहोस् ।
   (What types of sets are the sets P and Q — overlapping or disjoint sets? Write with reason.)[1K]
2. समूह P का सबै उपसमूहहरू लेख्नुहोस् ।
   (Write all subsets of set P.)[1U]
3. समूहहरू P र Q समूह U का कस्ता उपसमूहहरू हुन्, कारणसहित लेख्नुहोस् ।
   (What type of subsets of set U are the sets P and Q? Write with reason.)[1K]
1. From the Venn diagram, set \( P = \{x, y\} \) and set \( Q = \{w, z\} \).
   There is no overlapping region with common elements between \( P \) and \( Q \).
   Since \( P \cap Q = \emptyset \), sets \( P \) and \( Q \) are disjoint sets.
2. Set \( P = \{x, y\} \) has 2 elements.
   All subsets of \( P \) are:
   \( \emptyset, \{x\}, \{y\}, \{x, y\} \)
3. From the diagram:
   Universal set \( U = \{x, y, w, z, u, v\} \)
   \( P = \{x, y\} \subset U \) and \( Q = \{w, z\} \subset U \)
   Therefore, both \( P \) and \( Q \) are proper subsets of \( U \), because all their elements are in \( U \) and neither equals \( U \).
34. सर्वव्यापक समूह \( U = \) {6 भन्दा साना प्राकृतिक सङ्ख्याहरू} का दुईओटा उपसमूहहरू \( A =\) {6 भन्दा साना जोर सङ्ख्याहरू} र \( B =\) {6 भन्दा साना रूढ सङ्ख्याहरू} छन् ।
    (Two subsets of the universal set \( U = \{\text{Natural numbers less than } 6\} \) are \( A = \{\text{Even numbers less than } 6\} \) and \( B = \{\text{Prime numbers less than } 6\} \).)
1. समूहहरू U, A र B लाई सूचीकरण विधिमा लेख्नुहोस् ।
   (Write the sets U, A and B in listing method.)[1A]
2. माथिका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
   (Represent the above sets in Venn diagram.)[1U]
3. समूह A का सबै उपसमूहहरू लेख्नुहोस् ।
   (Write all subsets of set A.)[1U]
1. समूहहरू U, A र B लाई सूचीकरण विधिमा लेख्नुहोस् । (Write the sets U, A and B in listing method.)[1A]
   The sets are
   \( U = \{1, 2, 3, 4, 5\} \)
   \( A = \{2, 4\} \)
   \( B = \{2, 3, 5\} \)
2. माथिका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
   (Represent the above sets in Venn diagram.)[1U]
   The venn-diagram is
   
3. समूह A का सबै उपसमूहहरू लेख्नुहोस् ।
   (Write all subsets of set A.)[1U]
   Set \( A = \{2, 4\} \) has 2 elements, so it has \( 2^2 = 4 \) subsets.
   So, the subsets are
   \( \emptyset, \{2\}, \{4\}, \{2, 4\} \).
35. एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)
1. कुनै समूह र यसको अनुपयुक्त उपसमूहको सम्बन्ध कति हुन्छ ?
   (What is the relation of a set to its improper subset?)[1K]
2. समूह U का जम्मा कति ओटा उपसमूह निर्माण गर्न सकिन्छ ?
   (How many subsets of set U can be constructed?)[1U]
3. समूहहरू R र S लाई अलगिएका समूह बनाउन समूह S बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
   (Which elements of set S are to be removed to make the sets R and S disjoint sets?)[1HA]
1. कुनै समूह र यसको अनुपयुक्त उपसमूहको सम्बन्ध कति हुन्छ ? (What is the relation of a set to its improper subset?)[1K]
   An improper subset of a set is the set itself. Thus, the relation of a set to its improper subset is equality, i.e., if \( A \) is a set, its improper subset is \( A \), so \( A = A \).
2. समूह U का जम्मा कति ओटा उपसमूह निर्माण गर्न सकिन्छ ?
   (How many subsets of set U can be constructed?)[1U]
   From the Venn diagram, \( U = \{a, b, c, d, e, f, g\} \) has 7 elements. The number of subsets of a set with \( n \) elements is \( 2^n \).
   Thus, \( 2^7 = 128 \) subsets can be constructed from set \( U \).
3. समूहहरू R र S लाई अलगिएका समूह बनाउन समूह S बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
   (Which elements of set S are to be removed to make the sets R and S disjoint sets?)[1HA]
   From the Venn diagram, \( R = \{a, b, c\} \), \( S = \{b, c, d\} \), and their intersection is \( R \cap S = \{b, c\} \). To make \( R \) and \( S \) disjoint, \( b \) and \( c \) must be removed from \( S \).
36. सर्वव्यापक समूह \( U =\) {7 भन्दा साना पूर्ण सङ्ख्याहरू} का दुईओटा उपसमूहहरू \( A =\) {x : x एउटा विषम सङ्ख्या हो} र \( B =\) {y : y एउटा रूढ सङ्ख्या हो} छन् ।
    (Two subsets of the universal set \( U = \{\text{whole numbers less than } 7\} \) are \( A = \{x : x \text{ is an odd number}\} \) and \( B = \{y : y \text{ is a prime number}\} \).)
1. उपसमूहलाई परिभाषित गर्नुहोस् । Define subset.[1K]
2. माथि दिइएका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
   (Show the above sets in Venn diagram.)[1U]
3. समूहहरू A र B खण्डिएका वा अलगिएका कस्ता समूह हुन् ? कारणसहित लेख्नुहोस् ।
   (What type of sets are A and B — disjoint or overlapping? Write with reason.)[1HA]
1. A set \( X \) is called a subset of another set \( Y \) if every element of \( X \) is also an element of \( Y \).
   It is denoted by \( X \subseteq Y \).
2. Here
   Whole numbers less than 7: \( U = \{0, 1, 2, 3, 4, 5, 6\} \)
   Odd numbers: \( A = \{1, 3, 5\} \)
   Prime numbers: \( B = \{2, 3, 5\} \)
   Common elements: \( 3 \) and \( 5 \)
   The venn diagram is as below.
3. Since \( A \cap B = \{3, 5\} \neq \emptyset \), sets \( A \) and \( B \) are overlapping sets.

Test you understanding on Set (3 marks)