Set (Definition)

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।

---

---

---

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।

---

---

---

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 \(b \notin A\)

---

---

---

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

---

---

---

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.

---

---

---

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.

---

---

---

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?

---

---

---

---

---

---