Class 11 Maths Ncert Solutions Ex 1.4

Class 11 Maths Ncert Solutions Chapter 1 Ex 1.4

Q.1: Find the union of each of the following pairs of sets:

(i). P = {1, 4, 6} and Q = {1, 3, 4}

(ii). X = {a, e, i, o, u} and Y = {x, y, z}

(iii). X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 6}

(iv). X = {x: x is a natural number and 1 < x ≤ 5} and Y = {x: x is a natural number and 5 < x < 10}

(v). X = {4, 5, 6} and Y = Φ

Answer:

(i). 

Given:

P = {1, 4, 6} and Q = {1, 3, 4}

Therefore, P Q = {1, 3, 4, 6}

(ii). 

Given:

X = {a, e, i, o, u} and Y = {x, y, z}

Therefore, X Y = {x, y, z, a, e, i, o, u}

(iii).

Given:

X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 5}

X = {3, 6, 9 ….}

Y = {1, 2, 3, 4}

X Y = {1, 2, 3, 4, 6, 9 …. }

Therefore, X Y = {a:a = 1, 2, 3, 4 or multiple of 3}

(iv).

Given:

X = {x: x is a natural number and 1 < x ≤ 5} = {2, 3, 4, 5}

Y = {x: x is a natural number and 5 < x < 10} = {6, 7, 8, 9}

X Y = {2, 3, 4, 5, 6, 7, 8, 9}

Therefore, X Y = {a:a N and 1<a<10}

(v). 

Given:

X = {4, 5, 6} and Y = Φ

Therefore, X Y = {4, 5, 6}

 

 

Q.2:  Let A = {x, y} and B = {x, y, z}

Is A B? What is A B?

 

Answer:

Given:

A = {x, y} and B = {a, b, c}

Yes, A B that is A is subset of B.

Therefore, A B = {a, b, c} = B

 

 

Q.3: If X and Y are two sets such that X Y, then what is X Y?

 

Answer:

If X and Y are two sets such that X Y, then X Y = Y as all the elements of set X are present in set Y.

 

 

Q.4: If P = {1, 2, 3, 4}, Q = {3, 4, 5, 6}, R = {5, 6, 7, 8} and S = {7, 8, 9, 10}

Find the following:

(i).  P Q

(ii).  P R

(iii).  Q R

(iv).  Q S

(v).  P Q R

(vi).  P Q S

(vii).  Q R S

 

Answer:

Given:

P = {1, 2, 3, 4]

Q = {3, 4, 5, 6}

R= {5, 6, 7, 8} and,

S = {7, 8, 9, 10}

Now,

(i).  P Q = {1, 2, 3, 4, 5, 6}

(ii).  P R = {1, 2, 3, 4, 5, 6, 7, 8}

(iii).  Q R = {3, 4, 5, 6, 7, 8}

(iv).  Q S = {3, 4, 5, 6, 7, 8, 9, 10}

(v).  P Q R = {1, 2, 3, 4, 5, 6, 7, 8}

(vi).  P Q S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii).  Q R S = {3, 4, 5, 6, 7, 8, 9, 10}

 

 

Q.5:  Find the intersection of each of the following pairs of sets:

(i).  P = {1, 4, 6} and Q = {1, 3, 4}

(ii).  X = {a, e, i, o, u} and Y = {x, y, z}

(iii).  X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 6}

(iv).  X = {x: x is a natural number and 1 < x ≤ 5} and Y = {x: x is a natural number and 5 < x < 10}

(v).  X = {4, 5, 6} and Y = Φ

 

Answer:

(i).

Given:

P = {1, 4, 6} and Q = {1, 3, 4}

Therefore, P Q = {1, 3}

(ii). 

Given:

X = {a, e, i, o, u} and Y = {x, y, z}

Therefore, X Y = Φ

(iii).

Given:

X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 5}

X = {3, 6, 9 ….}

Y = {1, 2, 3, 4}

Therefore, X Y = {3}

(iv). 

Given:

X = {x: x is a natural number and 1 < x ≤ 5} = {2, 3, 4, 5}

Y = {x: x is a natural number and 5 < x < 10} = {6, 7, 8, 9}

Therefore, X Y = Φ

(v).  X = {4, 5, 6} and Y = Φ

Therefore, X Y = Φ

 

 

Q.6: If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}

Find the following:

(i).  A B

(ii).  B  C

(iii).  A  C  D

(iv).  A  C

(v).   B  D

(vi).  A  (B C)

(vii).  A  D

(viii).  A  (B D)

(ix).  (A  B)  (B C)

(x).  (A  D)  (B C)

 

Answer:

(i).  A B = {7, 9, 11}

(ii).  B C = {11, 13}

(iii).  A C  D

= {A C} D

= {11} {15, 17} = Φ

(iv).  A C = {11}

(v).  B D = Φ

(vi).  A (B C)

= (A B) (A C)

= {7, 9, 11} {11}

= {7, 9, 11}

(vii).  A D = Φ

(viii).  A (B D)

= (A B) (A D)

= (7, 9, 11} Φ = {7, 9, 11}

(ix).  (A B) (B C)

= {7, 9, 11} ) {7, 9, 11, 13, 15} = {7, 9, 11}

(x).  (A D) (B C)

= {3, 5, 7, 9, 11, 15, 17} ) {7, 9, 11, 13, 15} = {7, 9, 11, 15}

 

 

Q.7: If A = {y: y is a natural number}, B ={y: y is an even natural number}, C = {y: y is an odd natural number} and D = {y: y is a prime number}

Find the following:

(i). A B

(ii). A C

(iii). A D

(iv). B C

(v). B D

(vi). C D

 

Answer:

A = {y: y is a natural number} = {1, 2, 3, 4, 5….}

B = {y: y is an even natural number} = {2, 4, 6, 8 ….}

C = {y: y is an odd natural number} = {1, 3, 5, 7, 9 ….}

D = {y: y is a prime number} = {2, 3, 5, 7….}

Now,

(i).  A B = {y: y is a natural number} = B

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

(iii).  A D = {y: y is a prime number} = D

(iv).  B C = Φ

(v).  B D = {2}

(vi).  C D = {y: y is odd prime number}

 

 

Q.8: Which of the given pairs of sets are disjoint?

(i).  A = {1, 2, 3, 4} and B = {x: x is a natural number and 4 ≤ x ≤ 6}

(ii).  A = {a, e, i, o, u} and B = {c, d, e, f}

(iii).  A = {x: x is an even integer} and B = {x: x is an odd integer}

 

Answer:

(i).  A = {1, 2, 3, 4} and B = {x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}

A B = {1, 2, 3, 4} {4, 5 6} = {4}

Therefore, this pair of set is not disjoint.

(ii).  A = {a, e, i, o, u} and B = {c, d, e, f}

A B = {a, e, i, o, u} {c, d, e, f} = {e}

Therefore, this pair of set is not disjoint.

(iii).  A = {x: x is an even integer} and B = {x: x is an odd integer}

A B = {x: x is an even integer} {x: x is an odd integer} = Φ

Therefore, this pair of set is disjoint.

 

 

Q.9: If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}

Find the following:

(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

(xi).  C – D

(xii).  D – C

 

Answer:

(i).  A – B = {3, 6, 9, 15, 18, 21}

(ii).  A – C = {3, 9, 15, 18, 21}

(iii).  A – D = {3, 6, 9, 12, 18, 21}

(iv).  B – A = {4, 8, 16, 20}

(v).  C – A = {2, 4, 8, 10, 14, 16}

(vi).  D – A = {5, 10, 20}

(vii).  B – C = {20}

(viii).  B – D = {4, 8, 12, 16}

(ix).  C – B = {2, 6, 10, 14}

(x).  D – B = {5, 10, 15}

(xi).  C – D = {2, 4, 6, 8, 12, 14, 16}

(xii).  D – C = {5, 15, 20}

 

 

Q.10: If X = {a, b, c, d} and Y = {f, b, d, g}

Find the following:

(i).  X – Y

(ii).  Y – X

(iii).  X Y

 

Answer:

(i).  X – Y = {a, c}

(ii).  Y – X = {f, g}

(iii).  X Y = {b, d}

 

 

Q.11: What is R – Q, if R is the set of real numbers and Q is the set of rational?

 

Answer:

R: Set of real numbers

Q: Set of rational numbers

Hence, R – Q is a set of irrational numbers.

 

 

Q.12: State whether the following statements are true or false. Give reason.

(i).  A = {2, 3, 4, 5} and B = {3, 6} are disjoint sets.

(ii).  A = {a, e, i, o, u } and B = {a, b, c, d} are disjoint sets.

(iii).  A = {2, 6, 10, 14} and B = {3, 7, 11, 15} are disjoint sets.

(iv).  A = {2, 6, 10} and B = {3, 7, 11} are disjoint sets.

 

Answer:

(i).  False

Because, 3 {2, 3, 4, 5} and 3 {3, 6}

A B = {2, 3, 4, 5} {3, 6} = {3}

(ii). False

Because, a  {a, e, i, o, u} and a  {a, b, c, d}

A B = {a, e, i, o, u } {a, b, c, d} = {a}

(iii). True

Because, A B= {2, 6, 10, 14} {3, 7, 11, 15} = Φ

(iv). True

Because, A B = {2, 6, 10} {3, 7, 11} = Φ