Set Math 10 (e-Book)

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सिकाइ उपलब्धिः समुह -कक्षा १०

समूहका क्रियाहरू, भेनचित्र र गणनात्मकताको प्रयोग गरी तीनओटासम्म समूहसँग सम्बन्धित व्यावहारिक समस्याहरू समाधान गर्न

बिषयबस्तुः समुह -कक्षा १०

समूहहरूको संयोजन, प्रतिच्छेदन, पूरक र फरक क्रियाहरू प्रयोग हुने तीन समुहसम्मका दैनिक जीवनका व्यावहारिक समस्याहरूको समाधान (भेनचित्रको प्रयोसहित)

Specification Grid: समुह -कक्षा १०

S.N. Areas Working hours Knowledge Understanding Application Higher ability Total items questions Marks
ItemsMarks ItemsMarks ItemsMarks ItemsMarks
1. Sets 12 1 1 1 1 1 3 1 1 4 1 6

Question # 1

  1. [1]
  2. [1]
  3. [3]
  4. [1]

Problem Based on Two Sets

Learning Outcomes: दुईओटा समुहहरु सम्मिलित समस्याहरूको समाधान (भेनचित्रको प्रयोसहित)
Learning Activities

LA1: Set Operation

  1. Union of Set: समुह A र B को संयोजन (union) भनेको A वा B वा दुबै समुहका सबै सदस्यहरु समावेश भई बनेको हुन्छ। यसलाई AUB ले जनाईन्छ र "A संयोजन B" भनेर पढिन्छ।
    AUB = {x: x ∈ A or x ∈ B}.
  2. Intersection of Set: समुह A र B को प्रतिच्छेदन (intersection) भनेक A र B का सबै साझा सदस्यहरु मात्र समावेश भई बनेको हुन्छ। यसलाई A∩B ले जनाईन्छ र "A प्रतिच्छेदन B" भनेर पढिन्छ।
    A∩B = {x: x ∈ A and x ∈ B}.
  3. Difference of Set: समुह A र B को फरक (difference) भनेक A मा भएको तर B नभएको सबै सदस्यहरु समावेश भई बनेको हुन्छ। यसलाई A-B ले जनाईन्छ र "A फरक B" भनेर पढिन्छ।
    A-B = {x: x ∈ A and x ∉ B}.
  4. Complement of Set: समुह A को पुरक (complement) भनेको A मा नभएको सबै सदस्यहरु समावेश भई बनेको हुन्छ। यसलाई \(A' \text{ or} \overline{A} \text{ or} A^c\) ले जनाईन्छ र "U-A" भनेर पढिन्छ।
    \(\overline{A}\) = {x: x ∈ U and x ∉ A}

LA2: Cardinality of Set

The cardinality of a set A is the number of elements of the set A . The cardinality of a set A is usually denoted by n(A) but it can also be denoted as Card(A). For example:
  1. If \( A = \{x: x< 4, x \in \mathbb{W} \}\) then A = {0, 1, 2, 3} and n (A) = 4
  2. If B = { letters in the word “mathematics”} then B = {m, a, t, h, e, i, c, s} and n(B) = 8.

LA2: Set Operation and Cardinality

Study the given Venn-diagram, and find the elements and cardinality of tabulated sets.
SN Set Notation
1 \(A_o\) \(A_o=\{a,b\}\) or \(A-B=\{a,b\}\)
\(n_o(A)=2\) or \(n(A-B)=2\)		 \(A_o=\{1\}\) or \(A-B=\{1\}\)
\(n_o(A)=1\) or \(n(A-B)=1\)		 \(A_o=\{1,2,3\}\) or \(A-B=\{1,2,3\}\)
\(n_o(A)=3\) or \(n(A-B)=3\)
2 \(B_o\)
3 \(A \cap B\)
4 \(\overline{A \cup B}\)
5 \(A\)
6 \(\overline{A}\)
7 \(B\)
8 \(\overline{B}\)
9 \(A \triangle B\)
10 \(\overline{A \triangle B}\)
11 \(A \cup B\)
12 \(\overline{A _o}\)
13 \(\overline{B_o}\)
14 \(\overline{A \cap B}\)

LA3: Arithmetics on Cardinality for Problem Solving

Based on Venn-diagram with labeled cardinality, find cardinality of tabulated sets.
SN Set Notation
1 \(n_o(A)\)
who like only A

\(n_o(A)=p\)

\(n_o(A)=w\)

\(n_o(A)=a\)
2 \(n_o(B)\)
who like only B

\(n_o(B)=q\)

\(n_o(B)=0\)

\(n_o(B)=b\)
3 \(n(A \cap B)\)
who like A and B both

\(n(A \cap B)=r\)

\(n(A \cap B)=x\)

\(n(A \cap B)=0\)
4 \(n(\overline{A \cup B})\)
who like Neither A nor B

\(n(\overline{A \cup B})=s\)

\(n(\overline{A \cup B})=y\)

\(n(\overline{A \cup B})=c\)
5 \(n(A)\)
who like A

\(n(A)=p+r\)

\(n(A)=x+w\)

\(n(A)=a\)
6 \(n(\overline{A})\)
who does not like A

\(n(\overline{A})=q+s\)

\(n(\overline{A})=y\)

\(n(\overline{A})=b+c\)
7 \(n(B)\)
who like B

\(n(B)=q+r\)

\(n(B)=x\)

\(n(B)=b\)
8 \(n(\overline{B})\)
who does not like B

\(n(\overline{B})=p+s\)

\(n(\overline{B})=w+y\)

\(n(\overline{B})=a+c\)
9 \(n(A \triangle B)\)
who like exactly one

\(n(A \triangle B)=p+q\)

\(n(A \triangle B)=w\)

\(n(A \triangle B=a+b\)
10 \(n(\overline{A \triangle B})\)

\(n(\overline{A \triangle B})=r+s\)

\(n(\overline{A \triangle B})=x+y\)

\(n(\overline{A \triangle B})=c\)
11 \(n(A \cup B)\)
who like either A or B
who like at least one

\(n(A \cup B)=p+q+r\)

\(n(A \cup B)=x+w\)

\(n(A \cup B)=a+b\)
12 \(n(\overline{A _o})\)
Except who like A only

\(n(\overline{A _o})=q+r+s\)

\(n(\overline{A_o})=x+y\)

\(n(\overline{A_o})=b+c\)
13 \(n(\overline{B_o})\)
Except who like B only

\(n(\overline{B_o})=p+r+s\)

\(n(\overline{B_o})=w+x+y\)

\(n(\overline{B_o})=a+c\)
14 \(n(\overline{A \cap B})\)
Except who like both
who like at most one

\(n(\overline{A \cap B})=p+q+s\)

\(n(\overline{A \cap B})=w+y\)

\(n(\overline{A \cap B})=a+b+c\)

LA3: Problem Solving: Model 1

If cardinality of Set operations \(U, A_o,B_o,(A \cap B), (\overline{A \cup B})\) are given then use formula
\(n(U)=n_(A)+n_o(B)+n(A \cap B)+(\overline{A \cup B})\) and solve the following problems.

Sample Solution: Model 1

  1. Model 1
    100 जना मानिसहरूको समूहमा गरिएको सर्वेक्षणमा 50 जनाले आइफोन र 60 जनाले एन्ड्रोइडफोन मन पराएको पाइयो र 20 जनाले यी दुई फोनहरू मन पराएको पाइयो ।
    In a survey of 100 people, it was found that 50 people liked I-phone and 60 people like Android phone, and 20 people like both of these phones.
    1. आइफोन र एन्ड्रोइड दुबै फोन मन नपराउने मानिसहरूको गणनात्मकता समुह संकेतमा लेख्नुहोस ।
      Write the cardinality notation of people who like both of these phones. [1]
    2. माथिको जानकारीलाई भेनचित्रमा प्रस्तुत गर्नुहोस्।
      Present the above information in a Venn-diagram. [1]
    3. आइफोन र एन्ड्रोइड दुबै फोन मन नपराउने मानिसहरूक संख्या पत्ता लगाउनुहोस्।
      Find the number of people who does not like both of these phones. [3]
    4. आइफोन र एन्ड्रोइड फोन कुनै एउटा मात्र मन पराउनेको संख्या तुलना गर्नुहोस्।
      Compare the number of people who like only one of these two phones. [1]

    Solution 👉 Click Here

Questions: Model 1

Sample Solution: Model 2

Out of 100 students in a school, 60 passed in english, 70 in mathematics, 5 failed in both subjects and 10 did not appear in the examination.
एक विद्यालयका १०० जना विद्यार्थीमध्ये अंग्रेजीमा 60 जना उतीर्ण, गणितमा 70 जना उतीर्ण, र दुवै विषयमा 5 जना अनुत्तीर्ण भए र 10 जना परीक्षामा सहभागी भएनन् ।
  1. Write set notation of cardinality of students who passed in both subjects.
    दुवै विषयमा उत्तीर्ण विद्यार्थीहरूको संख्या समुह संकेतमा लेख्नुहोस् ।[1]
  2. Present above information in a Venn-diagram.
    माथिको जानकारी भेन-चित्रमा प्रस्तुत गर्नुहोस् ।[1]
  3. Find the number of students who passed in both subjects.
    दुवै विषयमा उत्तीर्ण भएका विद्यार्थीहरूको सङ्ख्या पत्ता लगाउनुहोस् ।[3]
  4. Find the ratio of passeed and failed students in both subjects.
    दुवै विषयमा उत्तीर्ण र अनुत्तीर्ण विद्यार्थीको अनुपात पत्ता लगाउनुहोस् ।[1]

Solution in English👉 Click Here

नेपालीमा उत्तर👉 क्लिक गर्नुहोस ।

Questions: Model 2

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