Types of sets
- 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
\(A = \{ x: x \text{ is a natural number less than } 1\}= \phi \) -
Singleton set
एउटा मात्र सदस्य (तत्व) भएको समूहलाई एकल समूह भनिन्छ।जस्तै
\(A = \{ x: x \text{ is even prime}\}= \{2\} \) -
Finite set
गन्ती गर्न सकिने सीमित संख्याहरु भएको समूहमा लाई finite set भनिन्छ।जस्तै
\(A = \{ \text{ whole numbers less than } 10\}= \{ 0,1,2,3,4,5,6,7,8,9\} \) -
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
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
A={समुद्रसँग जोडिएका राष्ट्रहरू}
B={भूपरिवेष्टित राष्ट्रहरू}
C={भारतसँग सिमा जोडिएका राष्ट्रहरू}
यि समूहको भेन चित्र Overlapping वा Disjoint के हुन्छ?
Overlapping Set
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
सूचीकरणविधि अनुसार
\(M=\)
\(N=\)
व्याख्या विधि अनुसार
\(M=\)
\(N=\)
\(M=\{3,,6,9,12,15\}\)
\(N=\{6,12,18,24\}\)
व्याख्या विधि अनुसार
\(M=\) set of the first five positive multiples of 3.
\(N=\) set of the first four positive multiples of 6.
U ={20 सम्मका 2 को अपवर्त्य हो ।}
P ={10 का गुणनखण्डहरू}
Q ={5 काअपवर्त्य हो ।} हो। र
R ={6 काअपवर्त्य 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.
- Set \( N \) is formed by adding 3 to each element of the set \( M \).
- Set \( P \) is formed by multiplying each element of set \( M \) by 2.
- Set \( Q \) is formed by collecting all the odd numbers of set \( M \).
- Set \( R \) is formed by collecting the multiples of 10 from set \( M \).
- Write the Universal set for the sets given above.
Given that
\( M = \{5, 10, 15, 20, 25, 30, 35\} \)
- Set \( N \) is formed by adding 3 to each element of \( M \), so
\( N = \{5+3, 10+3, 15+3, 20+3, 25+3, 30+3, 35+3\} = \{8, 13, 18, 23, 28, 33, 38\} \) - Set \( P \)is formed by multiplying each element of \( M \) by 2, so
\( P = \{2 \times 5, 2 \times 10, 2 \times 15, 2 \times 20, 2 \times 25, 2 \times 30, 2 \times 35\} = \{10, 20, 30, 40, 50, 60, 70\} \) - Set \( Q \)is formed by collecting all the odd numbers of \( M \), so
\( Q = \{5, 15, 25, 35\} \) - Set \( R \) is formed by collecting the multiples of 10 from \( M \), so
\( R = \{10, 20, 30\} \) - 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 \( R \) |
- \(A = \{\text{the set of even numbers up to } 10\}\)
- \(B = \{\text{the set of odd numbers up to } 10\}\)
- \(C = \{\text{the set of prime numbers up to } 10\}\)
- \(D = \{\text{multiples of 2}\}\).
Now, select the correct relationship for each pair:
| A | B | C | D | |
|---|---|---|---|---|
| A | ||||
| B | ||||
| C | ||||
| D |
- \(A=\) set of members of your family who likes bread as their breakfast.
- \(B=\) set of members of your family who likes other breakfast.
- Represent \(A\) and \(B\) in venn-diagram
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।
Equal Sets
Two or more sets are called equal (or identical or same) if they consist same elements. For example,
if \( A = \{1, 2\}\) and \( B = \{2, 1\}\) then \( A = B\) but the sets \( C = \{12\}\) and \( D =\{21\}\) are not equal.Equivalent Sets
Two sets \( A\) and \( B\) are called equivalent if their cardinal number is same, i.e., \( n (A) = n (B)\) . The symbol to denote equivalent sets is “\( \sim\) ”. For example,
if \( A = \{1, 2, 3\}\) and \( B = \{p, q, r\}\) then \( A \sim B\).Overlapping Sets
Two sets \( A\) and \( B\) are called overlapping set if they do have some common element. For example,
if \( A = \{1,2,3\}\) and \( B = \{3,4,5\}\) then \( A\) and \( B\) are overlapping sets as \( \{3\}\) is common to both sets \( A\) and \( B\) .Disjoint Sets
Two sets \( A\) and \( B\) are called disjoint if they have no elements in common. For example, if \( A = \{1, 2, 3\}\) and \( B = \{4,5, 6\}\) then \( A\) and \( B\) are disjoint as they have no element in common.
Comparable Sets
Two sets \( A\) and \( B\) are said to be comparable if \( A \subset B \) or \( B \subset A\) . For example, the sets \( A=\{a, b, c\}\) , and \( C=\{a, b, c, d\}\) are comparable. But the sets \( C=\{a, b, c\}\) and \( D=\{a, c, d, e\}\) are not comparable (incomparable) sets because C and D are overlapping sets.
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Comparable Sets
Two sets \( A\) and \( B\) are said to be comparable if \( A \subset B \) or \( B \subset A\) . For example, the sets \( A=\{a, b, c\}\) , and \( C=\{a, b, c, d\}\) are comparable. But the sets \( C=\{a, b, c\}\) and \( D=\{a, c, d, e\}\) are not comparable (incomparable) sets because C and D are overlapping sets.
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