NCERT Solutions class-11 Maths Exercise 1.5

Exercise 1.5

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)’

Ans. Given: 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 = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4}

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

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

= {1, 3, 5, 7, 9}

(iii) (A C)’ = U – (A C)

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({1, 2, 3, 4} {3, 4, 5, 6})

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 5, 6} = {7, 8, 9}

(iv) (A B)’ = U – (A B)

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({1, 2, 3, 4} {2, 4, 6, 8})

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 6, 8} = {5, 7, 9}

(v) (A’)’ = U – A’ = U – (U – A)

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4})

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {5, 6, 7, 8, 9}

= {1, 2, 3, 4} = A

(vi) (B – C)’ = U – (B – C)

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({2, 4, 6, 8} – {3, 4, 5, 6})

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 8}

= {1, 3, 4, 5, 6, 7, 9}

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2. If U = find the complement of the following sets:

(i) A =

(ii) B =

(iii) C =

(iv) D =

Ans. Given: U =

(i) A’ = U – A

=

(ii) B’ = U – B

=

(iii) C’ = U – C

=

(iv) D’ = U – D

=

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3. Taking the set of natural numbers as the universal set, write down the complement of the following set:

(i) { is an even natural number}

(ii) { is an odd natural number}

(iii) { is a positive multiple of 3}

(iv) { is a prime number}

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

(vi) { is a perfect square}

(vii) { is a perfect cube}

(viii) { + 5 = 8}

(ix) { +5=9}

(x)

(xi) { N and }

Ans. Given: U =

(i) Let A = { is an even natural number}

A’ = U – A = – { is an even natural number}

= { is an odd natural number}

(ii) Let A = { is an odd natural number}

A’ = U – A = – { is an odd natural number}

= { is an even natural number}

(iii) Let A = { is a positive multiple of 3}

A’ = U – A = – { is a positive multiple of 3}

= { is not a positive multiple of 3}

(iv) Let A = { is a prime number}

A’ = U – A = – { is a prime number}

= { is not a prime number}

(v) Let A = { is a natural number divisible by 3 and 5}

A’ = U – A = – { is a natural number divisible by 15}

= { is not divisible by 15}

(vi) Let A = { is a perfect square}

A’ = U – A = – { is a perfect square}

= { is not a perfect square}

(vii) Let A = { is a perfect cube}

A’ = U – A = – { is a perfect cube}

= { is not a perfect cube}

(viii) Let A = { + 5 = 8} = {3}

A’ = U – A = – {3}

= { 3}

(ix) Let A = { +5=9} = {2}

A’ = U – A = – {2}

= { 2}

(x) Let A = = {7, 8, 9, 10, ………}

A’ = U – A = – {7, 8, 9, 10, ………}

={1, 2, 3, 4, 5, 6} = { 7}

(xi) Let A = { N and } = {5, 6, 7, 8, ………..}

A’ = U – A = – {5, 6, 7, 8, ………..}

= {1, 2, 3, 4}

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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)

(ii)

Ans. Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},

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

(i) L.H.S. = = U – (A B)

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({2, 4, 6, 8} {2, 3, 5, 7})

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 4, 5, 6, 7, 8} = {1, 9}

R.H.S. = = (U – A) (U – B)

= ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}) ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 5, 7})

= {1, 3, 5, 7, 9} {1, 4, 6, 8, 9} = {1, 9}

L.H.S. = R. H. S.

(ii) L.H.S. =

= U – (A B)

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({2, 4, 6, 8} {2, 3, 5, 7})

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2}

= {1, 3, 4, 5, 6, 7, 8, 9}

R.H.S. = = (U – A) (U – B)

= ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}) ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 5, 7})

= {1, 3, 5, 7, 9} {1, 4, 6, 8, 9}

= {1, 3, 4, 5, 6, 7, 8, 9}

L.H.S. = R. H. S.

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5. Draw appropriate Venn diagrams for each of the following:

(i)

(ii)

(iii)

(iv)

Ans. (i) In the diagrams, shaded portion represents

(ii) In the diagrams, shaded portion represents

(iii) In the diagrams, shaded portion represents

(iv) In the diagrams, shaded portion represents

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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 what is A’?

Ans. Given: U = { is a triangle}

A = { is a triangle and has at least one angle different from }

A’ = U – A = { is a triangle and has all angles equal to }

= Set of all equilateral triangles

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

(i)

(ii)

(iii)

(iv)

Ans. (i)

(ii)

(iii)

(iv)