Class 11 Maths Ncert Solutions Ex 1.5

Class 11 Maths Ncert Solutions Chapter 1 Ex 1.5

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

X = {1, 2, 3, 4, 5}

Y = {2, 4, 6} and

Z = {2, 4, 5, 6}.

Find the following sets:

(i).   X’

(ii).  Y’

(iii).  (XY)

(iv).  (XZ)

(v).  (X)

(vi).  (YZ)

 

Answer:

Given:

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

X = {1, 2, 3, 4, 5}

Y = {2, 4, 6, 9} and

Z = {2, 4, 5, 6}

(i).  X’ = {6, 7, 8, 9}

(ii).  Y’ = {1, 3, 5, 7, 8, 9}

(iii).  (XY)

And, (XY)= {1, 2, 3, 4, 5, 6, 9}

Therefore, (XY)= {7, 8}

(iv).  (XZ)

And, (XZ) = {1, 2, 3, 4, 5, 6}

Therefore, (XZ) = {7, 8, 9}

(v).  (X) = X = {1, 2, 3, 4, 5}

(vi).  (YZ)

Since, (Y – Z) = {9}

Therefore, (Y – Z)’ = {1, 2, 3, 4, 5, 6, 7, 8}

 

 

Q.2: If U = {a, b, c, d, e, f, g, h}, find the complements of the given sets:

(i).  W = {a, b, c}

(ii).  X = {d, e, f, g}

(iii).  Y = {a, c, e, g}

(iv).  Z = {f, g, h, a}

 

Answer:

Given:

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

 (i).  W = {a, b, c}

W’ = {d, e, f, g, h}

(ii).  X = {d, e, f, g}

X’ = {a, b, c}

(iii).  Y = {a, c, e, g}

Y’ = {b, d, f}

(iv).  Z = {f, g, h, a}

Z’ = {b, c, d, e}

 

 

Q.3: Take natural numbers as the universal set. Write the complements of the given sets:

(i).  A = {y: y is an even natural number}

(ii).  B = {y: y is an odd natural number}

(iii).  C = {y: y is a positive multiple of 3}

(iv).  D = {y: y is a prime number}

(v).  E = {y: y is a natural number divisible by 3 and 5}

(vi).  F = {y: y is a perfect square}

(vii).  G = {y: y is perfect cube}

(viii).  H = {y: y + 5 = 8}

(ix).  I = {y: 2y + 5 = 9}

(x).  J = {y: y 7}

(xi).  K = {y: y N and 2y + 1 > 10}

Answer:

Given:

U = set of natural numbers = N

(i).  A’ = {y: y is an even natural number}’

= {y: y is an odd natural number}

 

(ii).  B’ = {y: y is an odd natural number}

= {y: y is an even natural number}

 

(iii).  C’ = {y: y is a positive multiple of 3}’

= {y: y N and y is not a multiple of 3}

 

(iv).  D’ = {y: y is a prime number}’

= {y: y is a positive composite number and y = 1}

 

(v).  E’ = {y: y is a natural number divisible by 3 and 5}’

= {y: y is a natural number that is not divisible by 3 or 5}

 

(vi).  F’ = {y: y is a perfect square}’

= {y: y N and y is not a perfect square}

 

(vii).  G’ = {y: y is perfect cube}’

= {y: y N and y is not a perfect cube} 

 

(viii).  H’ = {y: y + 5 = 8}’

= {y: y N and y 3}

 

(ix).  I’ = {y: 2y + 5 = 9}’

= {y: y N and y 2}

 

(x).  J’ = {y: y 7}’

= {y: y N and x < 7}

 

(xi).  K’ = {y: y N and 2y + 1 > 10}

= {y: y N and y 92}

 

 

Q.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).  (AB) = AB

(ii).  (AB) = AB

Answer:

Given:

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

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

(i).  (AB)

= {2, 3, 4, 5, 6, 7, 8}’ = {1, 9}

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

Therefore, (AB) = AB

(ii).  (AB)

= {2}’ = {1, 3, 4, 5, 6, 7, 8, 9}

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

Therefore, (AB) = AB

 

 

Q.5: Draw the Venn diagrams for the following:

(i).  (AB)

(ii).  AB

(iii).  (AB)

(iv).  AB

Answer:

(i).  (AB):

1

(ii).  AB:

1

(iii).  (AB):

11

(iv).  AB:

11

 

 

Q.6: Let U be the universal set that is the set of all triangles in a plane. If X is the set of all triangles with at least one angle different from 60, what is X’ ?

 

Answer:

X’ is the set of all equilateral triangles.

Q.7: Complete the given statements using proper symbols:

(i).  AA

(ii).  Ø A

(iii).  AA

(iv).  UA

 

Answer:

(i).  AA = U

(ii).  Ø A = UA = A

(iii).  AA = Ø

(iv).  UA = Ø A = Ø