ap ssc sets exercise 2
Exercise 2
Do this
1. Let A = {1, 3, 7, 8} and B = {2, 4, 7, 9}. Find A ∩ B.
A ∩ B={1, 3, 7, 8} ∩ {2, 4, 7, 9} ={7}
2. If A = {6, 9, 11}; B = { }, find A ∪ 𝜙
A ∪ 𝜙 = {6, 9, 11} ∪ { } = {6, 9, 11}
3. A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; B = {2, 3, 5, 7}. Find A ∩ B and show that A ∩ B = B.
A ∩ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∩ {2, 3, 5, 7} = {2, 3, 5, 7} = B
4. If A = {4, 5, 6}; B = {7, 8} then show that A ∪ B = B ∪ A.
A ∪ B = {4, 5, 6} ∪ {7, 8} = { 4,5,6,7,8}
B ∪ A = {7, 8} ∪ {4, 5, 6} = { 4,5,6,7,8}
Try this
1.List out some sets A and B and choose their elements such that A and B are disjoint
A= { 1,2}, B={3,4,5}
A= {𝑎,𝑏,𝑐},and B= {𝑥,𝑦,𝑧}
2. If A = {2, 3, 5}, find A ∪ 𝜙 and 𝜙 ∪ A and compare.
A ∪ 𝜙 = {2, 3, 5} ∪ { } = {2, 3, 5}
𝜙 ∪ A = { } ∪ {2, 3, 5} = {2, 3, 5}
A ∪ 𝜙 = 𝜙 ∪ A = A
3. If A = {1, 2, 3, 4}; B = {1, 2, 3, 4, 5, 6, 7, 8} then find A ∪ B, A ∩ B. What do you notice about the result?
A ∪ B = {1, 2, 3, 4} ∪ {1, 2, 3, 4, 5, 6, 7, 8}={1, 2, 3, 4, 5, 6, 7, 8}=B
A ∩ B ={1, 2, 3, 4} ∩ {1, 2, 3, 4, 5, 6, 7, 8}= {1, 2, 3, 4}=A
If A ⊂ B then A ∪ B = B and A ∩ B = A
4. A = {1, 2, 3, 4, 5, 6}; B = {2, 4, 6, 8, 10}. Find the intersection of A and B.
Think-Discuss
The intersection of any two disjoint sets is a null set. Justify your answer.
If A={4,5} and B= {2,3,6} these sets are disjoint sets.There is no common element in both. so Intersection of disjoint sets is a null set.
i.e. A ∩ B ={}=𝜙
Do this
1. If A = {1, 2, 3, 4 ,5}; B = {4, 5, 6, 7}then find A – B and B – A. Are they equal?
A – B = {1, 2, 3, 4 ,5} – {4, 5, 6, 7} = {1,2,3 }
B – A = {4, 5, 6, 7} – {1, 2, 3, 4 ,5} ={6,7 }
2. If V = {a, e, i, o, u} and B = {a, i,k, u}, find V – B and B – V.
V – B = {a, e, i, o, u} – {a, i,k, u} ={ e,o }
B – V = {a, i,k, u} – {a, e, i, o, u} ={ k }
Think-Discuss
The sets A – B, B – A and A ∩ B are mutually disjoint sets. Use examples to observe if this is true.
If A = {1, 2, 3, 4 ,5}; B = {4, 5, 6, 7}then
A – B = {1,2,3 } and B – A = {6,7 }
A ∩ B = {1, 2, 3, 4 ,5} ∩ {4, 5, 6, 7} = { 4,5}
There is no common elements in A – B, B – A and A ∩ B so they are mutually disjoint sets.
1. If A = {1, 2, 3, 4}; B = {1, 2, 3, 5, 6} then find A ∩ B and B ∩ A. Are they equal?
A ∩ B = {1, 2, 3, 4} ∩ {1, 2, 3, 5, 6} = {1,2,3}
B ∩ A = {1, 2, 3, 5, 6} ∩ {1, 2, 3, 4} = {1,2,3}
`therefore`A ∩ B = B ∩ A
2. A = {0, 2, 4}, find A ∩ 𝜙 and A ∩ A. Comment.
A ∩ 𝜙 = {0, 2, 4} ∩ {} = {} = 𝜙
A ∩ A = {0, 2, 4} ∩ {0, 2, 4} = {0, 2, 4} = A
A ∩ 𝜙 = 𝜙 and A ∩ A= A
3. If A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}, find A – B and B – A.
A – B = {2, 4, 6, 8, 10} – {3, 6, 9, 12, 15} = {2,4,8,10}
B – A = {3, 6, 9, 12, 15} – {2, 4, 6, 8, 10} = {3,9,12,15}
4. If A and B are two sets such that A ⊂ B then what is A ∪ B?
If A ⊂ B , all elements of A is also in B so A ∪ B = B
5. If A = {x : x is a natural number} B = {x : x is an even natural number} C = {x : x is an odd natural number} D = {x : x is a prime number}
Find A ∩ B, A ∩ C, A ∩ D, B ∩ C, B ∩ D, C ∩ D.
A = {x : x is a natural number} = {1,2,3,4,5,----}
B = {x : x is an even natural number}={2,4,6,8,---}
C = {x : x is an odd natural number} ={1,3,5,7,9,---}
D = {x : x is a prime number}={2,3,5,7,11,---}
A ∩ B= {1,2,3,4,5,----} ∩ {2,4,6,8,---}
= {2,4,6,8,---}= B
A ∩ C = {1,2,3,4,5,----} ∩ {1,3,5,7,9,---}
= {1,3,5,7,9,---}=C
A ∩ D = {1,2,3,4,5,----} ∩ {2,3,5,7,11,---}
={2,3,5,7,11,---}=D
B ∩ C = {2,4,6,8,---}∩ {1,3,5,7,9,---}
= { } = 𝜙
B ∩ D = {2,4,6,8,---} ∩ {2,3,5,7,11,---}
= {2}
C ∩ D = {1,3,5,7,9,---} ∩ {2,3,5,7,11,---}
= {3,5,7,11,13,---}
6. If A = {3, 6, 9, 12, 15, 18, 21}; B = {4, 8, 12, 16, 20} C = {2, 4, 6, 8, 10, 12, 14, 16}; D = {5, 10, 15, 20} find
(i) A – B (ii) A – C (iii) A – D (iv) B – A (v) C – A (vi) D – A (vii) B – C (viii) B – D (ix) C – B (x) D – B
(i) A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20}
={3,6,9,15,18,21}
(ii) A – C ={3, 6, 9, 12, 15, 18, 21} –{2, 4, 6, 8, 10, 12, 14, 16}
={3,9,15,18,21}
(iii) A – D ={3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20}
={3,6,9,12,18,21 }
(iv) B – A={4, 8, 12, 16, 20} –{3, 6, 9, 12, 15, 18, 21}
={4,8,16,20}
(v) C – A ={2, 4, 6, 8, 10, 12, 14, 16}– {3, 6, 9, 12, 15, 18, 21}
={2,4,8,10,14,16}
(vi) D – A ={5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21}
={5,10,20}
(vii) B – C ={4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12, 14, 16}
={20}
(viii) B – D ={4, 8, 12, 16, 20} – {5, 10, 15, 20}
={4,8,12,16}
(ix) C – B={2, 4, 6, 8, 10, 12, 14, 16}– {4, 8, 12, 16, 20}
={2,6,10,14}
(x) D – B={5, 10, 15, 20} – {4, 8, 12, 16, 20}
={5,10,15}
7. State whether each of the following statement is true or false. Justify you answers.
(i) 2, 3, 4, 5} and {3, 6} are disjoint sets.
False, because there is a common element ‘3’ in given sets.
(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
False, because there is a common element ‘a’ in given sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
True, because there are no common elements in given sets.
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.
True, because there are no common elements in given sets.
❮ Equal sets
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