NCERT Solutions for Class 11 Maths Chapter 1 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 $\cup$ C)′

(iv) (A $\cup$ B)′

(v) (A')'

(vi) (B – C)'

Answer:

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 = {5,6,7,8,9}

(ii) B′ = U - B = {1,3,5,7,9}

(iii) A $\cup$ C = {1,2,3,4,5,6}

(A $\cup$ C)′ = U - (A $\cup$ C) = {7,8,9}

(iv) (A $\cup$ B) = {1,2,3,4,6,8}

(A $\cup$ B)′ = U - (A $\cup$ B) = {5,7,9}

(v) (A')' = A = { 1, 2, 3, 4}

(vi) (B – C) = {2,8}

(B – C)' = U - (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}

Answer:

U = { a, b, c, d, e, f, g, h}

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

A' = U - A = {d,e,f,g,h}

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

B' = U - B = {a,b,c,h}

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

C' = U - C = {b,d,f,h}

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

D' = U - D = {b,c,d,e}

Question 3: 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}

Answer:

Universal set = U = {1,2,3,4,5,6,7....................}

(i) {x : x is an even natural number} = {2,4,6,8,..........}

{x : x is an even natural number}'= U - {x : x is an even natural number} = {1,3,5,7,9,..........} = {x : x is an odd natural number}

(ii) { x : x is an odd natural number }' = U - { x : x is an odd natural number } = {x : x is an even natural number}

(iii) {x : x is a positive multiple of 3}' = U - {x : x is a positive multiple of 3} = {x : x , x $\in$ N and is not a positive multiple of 3}

(iv) { x : x is a prime number }' = U - { x : x is a prime number } = { x : x is a positive composite number and 1 }

(v) {x : x is a natural number divisible by 3 and 5}' = U - {x : x is a natural number divisible by 3 and 5} = {x : x is a natural number not divisible by 3 or 5}

Question 3: Taking the set of natural numbers as the universal set, write down the complements of the following sets:

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

(vii) { x : x is a perfect cube}

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

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

(x) { x : x $\ge$ 7 }

(xi) { x : x $\in$N and 2x + 1 $>$ 10 }

Answer:

Universal set = U = {1,2,3,4,5,6,7,8.............}

(vi) { x : x is a perfect square }' = U - { x : x is a perfect square } = { x : x $\in$ N and x is not a perfect square }

(vii) { x : x is a perfect cube}' = U - { x : x is a perfect cube} = { x : x $\in$ N and x is not a perfect cube}

(viii) { x : x + 5 $=$8 }' = U - { x : x + 5 $=$8 } = U - {3} = { x : x$\in$ N and x $\ne$ 3 }

(ix) { x : 2x + 5 $=$9}' = U - { x : 2x + 5 $=$9} = U -{2} = { x : x$\in$ N and x $\ne$ 2}

(x) { x : x $\ge$ 7 }' = U - { x : x $\ge$ 7 } = { x : x$\in$ N and x $<$ 7 }

(xi) { x : x $\in$N and 2x + 1 $>$ 10 }' = U - { x : x $\in$N and x $>$ 9/2 } = { x : x$\in$ N and x $\le$ 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 $\cup$ B)′ = A′ $\cap$ B′

(ii) (A $\cap$ B)′ = A′ $\cup$ B′

Answer:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}

(i) (A $\cup$ B)′ = A′ $\cap$ B′

L.H.S = (A $\cup$ B)′ = U - (A $\cup$ B) = {1,9}

R.H.S = A′ $\cap$ B′ = {1,3,5,7,9} $\cap$ {1,4,6,8,9} = {1,9}

L.H.S = R.H.S

Hence,the statement is true.

(ii) (A $\cap$ B)′ = A′ $\cup$ B′

L.H.S = U - (A $\cap$ B) ={1,3,4,5,6,7,8,9}

R.H.S = A′ $\cup$ B′ = {1,3,5,7,9} $\cup$ {1,4,6,8,9} = {1,3,4,5,6,7,8,9}

L.H.S = R.H.S

Hence,the statement is true.

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

(i) (A ∪ B)′

(ii) A′ ∩ B′

(iii) (A ∩ B)′

(iv) A′ ∪ B′

Answer:

(i) (A ∪ B)′

(A ∪ B) is in yellow colour

(A ∪ B)′ is in green colour

ii) A′ ∩ B′ is represented by the green colour in the below figure

iii) (A ∩ B)′ is represented by green colour in the below diagram and white colour represents (A ∩ B)

iv) A′ ∪ B′

The green colour represents 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 ${60}^{\circ }$, what is A′?

Answer:

A' is the set of all triangles whose angle is ${60}^{\circ }$ in other words A' is set of all equilateral triangles.

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

(i) A $\cup$ A′ $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(ii) ${\varphi }^{\prime }$ $\cap$A $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(iii) A $\cap$ A′ $=$$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(iv) U′ $\cap$ A $=$$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Answer:

The following are the answers for the questions

(i) A $\cup$ A′ $=$ U

(ii) $\varphi$′ $\cap$ A $=$ A

(iii) A $\cap$ A′ $=$ $\varphi$

(iv) U′ $\cap$ A $=$ $\varphi$