TOPICS

Exercise - 1.5

Sets

**Question-1 :-** Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find

(i) A'

(ii) B'

(iii) (A ∪ C)'

(iv) (A ∪ B)'

(v) (A')'

(vi) (B - C)'

Given that U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}; (i) A' = U - A A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4} A' = {5, 6, 7, 8, 9}

(ii) B' = U - B B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8} B' = {1, 3, 5, 7, 9}

(iii) (A ∪ C)' = U - (A ∪ C) A ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6} A ∪ C = {1, 2, 3, 4, 5, 6} (A ∪ C)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6} (A ∪ C)' = {7, 8, 9}

(iv) (A ∪ B)' = U - (A ∪ B) A ∪ B = {1, 2, 3, 4} ∪ {2, 4, 6, 8} A ∪ B = {1, 2, 3, 4, 6, 8} (A ∪ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 6, 8} (A ∪ B)' = {5, 7, 9}

(v) (A')' = U - A' A' = U - A A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4} A' = {5, 6, 7, 8, 9} (A')' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {5, 6, 7, 8, 9} (A')' = {1, 2, 3, 4} = A

(vi) (B - C)' = U - (B - C) B - C = {2, 4, 6, 8} - {3, 4, 5, 6} B - C = {2, 8} (B - C)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 8} (B - C)' = {1, 3, 4, 5, 6, 7, 9}

**Question-2 :-** If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:

(i) A = {a, b, c}

(ii) B = {d, e, f, g}

(iii) C = {a, c, e, g}

(iv) D = {f, g, h, a}

Given that U = {a, b, c, d, e, f, g, h}. (i) A = {a, b, c} A' = U - A A' = {a, b, c, d, e, f, g, h} - {a, b, c} A' = {d, e, f, g, h}

(ii) B = {d, e, f, g} B' = U - B B' = {a, b, c, d, e, f, g, h} - {d, e, f, g} B' = {a, b, c, h}

(iii) C = {a, c, e, g} C' = U - C C' = {a, b, c, d, e, f, g, h} - {a, c, e, g} C' = {b, d, f, h}

(iv) D = {f, g, h, a} D' = U - D D' = {a, b, c, d, e, f, g, h} - {f, g, h a} D' = {b, c, d, e}

**Question-4 :-** Taking the set of natural numbers as the universal set, write down the complements of the following sets:

(i) {x: x is an even natural number}

(ii) {x: x is an odd natural number}

(iii) {x: x is a positive multiple of 3}

(iv) {x: x is a prime number}

(v) {x: x is a natural number divisible by 3 and 5}

(vi) {x: x is a perfect square}

(vii) {x: x is perfect cube}

(viii) {x: x + 5 = 8}

(ix) {x: 2x + 5 = 9}

(x) {x: x ≥ 7}

(xi) {x: x ∈ N and 2x + 1 > 10}

Given that U = {x : x is set of natural numbers} = {1, 2, 3,....} (i) A = {x: x is an even natural number} A = {2, 4, 6,....} A' = U - A A' = {1, 2, 3,....} - {2, 4, 6,....} A' = {1, 3, 5....} A' = {x : x is set of odd natural numbers, x ∈ N}

(ii) B = {x: x is an odd natural number} B = {1, 3, 5,....} B' = U - B B' = {1, 2, 3,....} - {1, 3, 5,....} B' = {2, 4, 6....} B' = {x : x is set of even natural numbers, x ∈ N}

(iii) C = {x: x is a positive multiple of 3} C = {3, 6, 9,....} C' = U - C C' = {1, 2, 3,....} - {3, 6, 9,....} C' = {2, 4, 5....} C' = {x : x is not a positive multiple of 3, x ∈ N}

(iv) D = {x: x is a prime number} D = {2, 3, 5,....} D' = U - D D' = {1, 2, 3,....} - {2, 3, 5,....} D' = {1, 4, 6....} D' = {x : x is not a prime number, x ∈ N}

(v) E = {x: x is a natural number divisible by 3 and 5} E = {15, 30, 45,....} E' = U - E E' = {1, 2, 3,....} - {15, 30, 45,....} E' = {1, 3, 5....} E' = {x : x is not a natural number divisible by 3 and 5, x ∈ N}

(vi) F = {x: x is a perfect square} F = {1, 4, 9,....} F' = U - F F' = {1, 2, 3,....} - {1, 4, 9,....} F' = {2, 3, 5....} F' = {x : x is not a perfect square, x ∈ N}

(vii) G = {x: x is perfect cube} G = {1, 8, 27,....} G' = U - G G' = {1, 2, 3,....} - {1, 8, 27,....} G' = {2, 3, 4....} G' = {x : x is not a perfect cube, x ∈ N}

(viii) H = {x: x + 5 = 8} x + 5 = 8 x = 8 - 5 x = 3 H = {3} H' = U - H H' = {1, 2, 3,....} - {3} H' = {1, 2, 4....} H' = {x : x ≠ 3, x ∈ N}

(ix) I = {x: 2x + 5 = 9} 2x + 5 = 9 2x = 9 - 5 2x = 4 x = 4/2 x = 2 I = {2} I' = U - I I' = {1, 2, 3,....} - {2} I' = {1, 3, 4,....} I' = {x : x ≠ 2, x ∈ N}

(x) J = {x: x ≥ 7} J = {7, 8, 9,....} J' = U - J J' = {1, 2, 3,....} - {7, 8, 9,....} J' = {1, 2, 3, 4, 5, 6} J' = {x : x < 7, x ∈ N}

(xi) K = {x: x ∈ N and 2x + 1 > 10} K = {5, 6, 7,....} K' = U - K K' = {1, 2, 3,....} - {5, 6, 7,....} k' = {1, 2, 3, 4} K' = {x: x ∈ N and x ≤ 9/2}

**Question-4 :-** If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that

(i) (A ∪ B)' = A'∩ B'

(ii) (A ∩ B)' = A' ∪ B'

Given that U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. (i) (A ∪ B)' = A'∩ B' L.H.S (A ∪ B)' = U - (A ∪ B) A ∪ B = {2, 4, 6, 8} ∪ {2, 3, 5, 7} A ∪ B = {2, 3, 4, 5, 6, 7, 8} (A ∪ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 4, 5, 6, 7, 8} (A ∪ B)' = {1, 9} R.H.S A'∩ B' = (U - A) ∩ (U - B) A' = U - A A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8} A' = {1, 3, 5, 7, 9} B' = U - B B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7} B' = {1, 4, 6, 8, 9} A'∩ B' = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} A'∩ B' = {1, 9} L.H.S = R.H.S

(ii) (A ∩ B)' = A' ∪ B' L.H.S (A ∩ B)' = U - (A ∩ B) A ∩ B = {2, 4, 6, 8} ∩ {2, 3, 5, 7} A ∩ B = {2} (A ∩ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2} (A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9} R.H.S A'∪ B' = (U - A) ∪ (U - B) A' = U - A A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8} A' = {1, 3, 5, 7, 9} B' = U - B B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7} B' = {1, 4, 6, 8, 9} A'∪ B' = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} A'∪ B' = {1, 3, 4, 5, 6, 7, 8, 9} L.H.S = R.H.S

**Question-5 :-** Draw appropriate Venn diagram for each of the following:

(i) (A ∪ B)'

(ii) A'∩ B'

(iii) (A ∩ B)'

(iv) A' ∪ B'

(i) (A ∪ B)'

(ii) A'∩ B'

(iii) (A ∩ B)'

(iv) A' ∪ B'

**Question-6 :-** Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is ?

Given that U = {x : x is a triangle}. A = {x : x is a triangle and has at least one angle different from 60°} A' = U - A A' = {x : x is a triangle} - {x : x is a triangle and has at least one angle different from 60°} A' = {x : x is a triangle and has all angles equal to 60°} A' = set of all equilateral triangles

**Question-7 :-** Fill in the blanks to make each of the following a true statement:

(i) A ∪ A'= ...

(ii) Φ′∩ A = ...

(iii) A ∩ A' = ...

(iv) U'∩ A' = ...

(i) A ∪ A'= U (ii) Φ′∩ A = U ∩ A = A (iii) A ∩ A' = Φ (iv) U'∩ A' = Φ ∩ A = Φ

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