NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.4 - Sets

Question 1:(i) Find the union of each of the following pairs of sets :

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

Answer:

Union of X and Y is X $\cup$ Y = {1,2,3,5}

Question:1(ii) Find the union of each of the following pairs of sets :

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

Answer:

Union of A and B is A $\cup$ B = {a,b,c,e,i,o,u}.

Question 1:(iii) Find the union of each of the following pairs of sets :

A = {x : x is a natural number and multiple of 3} B = {x : x is a natural number less than 6}

Answer:

Here ,

A = {3,6,9,12,15,18,............}

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

Union of A and B is A $\cup$ B.

A $\cup$ B = {1,2,3,4,5,6,9,12,15........}

Question 1:(iv) Find the union of each of the following pairs of sets :

A = {x : x is a natural number and 1 $<$ x $\le$6 }

B = {x : x is a natural number and 6 $<$ x $<$10 }

Answer:

Here,

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

B = {7,8,9}

A $\cup$ B = {2,3,4,5,6,7,8,9}

or it can be written as A $\cup$ B = {x : x is a natural number and 1$<$ x $<$10 }

Question 1:(v) Find the union of each of the following pairs of sets :

A = {1, 2, 3} B = $\varphi$

Answer:

Here,

A union B is A $\cup$ B.

A $\cup$ B = {1,2,3}

Question 2: Let A = { a, b }, B = {a, b, c}. Is A $\subset$ B ? What is A $\cup$ B ?

Answer:

Here,

We can see elements of A lie in set B.

Hence, A $\subset$ B.

And, A $\cup$ B = {a,b,c} = B

Question 3: If A and B are two sets such that A $\subset$ B, then what is A $\cup$ B ?

Answer:

If A is subset of B then A $\cup$ B will be set B.

Question 4:(i) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

A $\cup$ B

Answer:

Here,

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

The union of the set can be written as follows

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

Question 4:(ii) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

A $\cup$ C

Answer:

Here,

A = {1, 2, 3, 4}

C = {5, 6, 7, 8 }

The union can be written as follows

A $\cup$ C = {1,2,3,4,5,6,7,8}

Question 4:(iii) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

B $\cup$ C

Answer:

Here,

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

C = {5, 6, 7, 8 }

The union of the given sets are

B $\cup$ C = { 3,4,5,6,7,8}

Question 4:(iv) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

B $\cup$ D

Answer:

Here,

B = {3, 4, 5, 6}

D = { 7, 8, 9, 10 }

B $\cup$ D = {3,4,5,6,7,8,9,10}

Question 4:(v) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

A $\cup$ B $\cup$ C

Answer:

Here,

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

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

C = {5, 6, 7, 8 }

The union can be written as

A $\cup$ B $\cup$ C = {1,2,3,4,5,6,7,8}

Question 4:(vi) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

A $\cup$ B $\cup$ D

Answer:

Here,

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

B = {3, 4, 5, 6}

D = { 7, 8, 9, 10 }

The union can be written as

A $\cup$ B $\cup$ D = {1,2,3,4,5,6,7,8,9,10}

Question 4:(vii) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

B $\cup$ C $\cup$ D

Answer:

Here,B = {3, 4, 5, 6},

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

D = { 7, 8, 9, 10 }

The union can be written as

B $\cup$ C $\cup$ D = {3,4,5,6,7,8,9,10}

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

(i) A $\cap$ B (ii) B $\cap$ C

(iii) A $\cap$C $\cap$ D (iv) A $\cap$ C

(v) B $\cap$ D (vi) A $\cap$ (B $\cup$ C)

(vii) A $\cap$ D (viii) A $\cap$ (B $\cup$ D)

(ix) ( A $\cap$ B ) $\cap$ ( B $\cup$ C ) (x) ( A $\cup$ D) $\cap$ ( B $\cup$ C)

Answer:

Here, A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}

(i) A $\cap$ B = {7,9,11} (vi) A $\cap$ (B $\cup$ C) = {7,9,11}

(ii) B $\cap$ C = { 11,13} (vii) A $\cap$ D = $\varphi$

(iii) A $\cap$C $\cap$ D = $\varphi$ (viii) A $\cap$ (B $\cup$ D) = {7,9,11}

(iv) A $\cap$ C = { 11 } (ix) ( A $\cap$ B ) $\cap$ ( B $\cup$ C ) = {7,9,11}

(v) B $\cap$ D = $\varphi$ (x) ( A $\cup$ D) $\cap$ ( B $\cup$ C) = {7,9,11,15}

Question 7: If A = {x : x is a natural number }, B = {x : x is an even natural number} C = {x : x is an odd natural number} and D = {x : x is a prime number }, find

(i) A $\cap$ B

(ii) A $\cap$ C

(iii) A $\cap$ D

(iv) B $\cap$ C

(v) B $\cap$ D

(vi) C $\cap$ D

Answer:

Here, A = {1,2,3,4,5,6...........}

B = {2,4,6,8,10...........}

C = {1,3,5,7,9,11,...........}

D = {2,3,5,7,11,13,17,......}

(i) A $\cap$ B = {2,4,6,8,10........} = B

(ii) A $\cap$ C = {1,3,5,7,9.........} = C

(iii) A $\cap$ D = {2,3,5,7,11,13.............} = D

(iv) B $\cap$ C = $\varphi$

(v) B $\cap$ D = {2}

(vi) C $\cap$ D = {3,5,7,11,13,..........} = $(x:xisoddprimenumber)$

Question 8:(i) Which of the following pairs of sets are disjoint

{1, 2, 3, 4} and {x : x is a natural number and 4 $\le$x $\le$6 }

Answer:

Here, {1, 2, 3, 4} and {4,5,6}

{1, 2, 3, 4} $\cap$ {4,5,6} = {4}

Hence,it is not a disjoint set.

Question 8:(ii) Which of the following pairs of sets are disjoint

{ a, e, i, o, u } and { c, d, e, f }

Answer:

Here, { a, e, i, o, u } and { c, d, e, f }

{ a, e, i, o, u } $\cap$ { c, d, e, f } = {e}

Hence,it is not disjoint set.

Question 8:(iii) Which of the following pairs of sets are disjoint

{x : x is an even integer } and {x : x is an odd integer}

Answer:

Here, {x : x is an even integer } and {x : x is an odd integer}

{2,4,6,8,10,..........} and {1,3,5,7,9,11,.....}

{2,4,6,8,10,..........} $\cap$ {1,3,5,7,9,11,.....} = $\varphi$

Hence,it is disjoint set.

Question 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 }, D = {5, 10, 15, 20 }; find

(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:

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

The given operations are done as follows

(i) A – B = {3,6,9,15,18,21} (vii) B – C = {20}

(ii) A – C = {3,9,15,18,21} (viii) B – D = {4,8,12,16}

(iii) A – D = {3,6,9,12,18,21} (ix) C – B = {2,6,10,14}

(iv) B – A = {4,8,16,20} (x) D – B = {5,10,15}

(v) C – A = {2,4,8,10,14,16} (xi) C – D = {2,4,6,8,12,14,16}

(vi) D – A = {5,10,20} (xii) D – C = {5,15,20}

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

(i) X – Y

(ii) Y – X

(iii) X $\cap$ Y

Answer:

X= { a, b, c, d } and Y = { f, b, d, g}

(i) X – Y = {a,c}

(ii) Y – X = {f,g}

(iii) X $\cap$ Y = {b,d}

Question 11: If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

Answer:

R = set of real numbers.

Q = set of rational numbers.

R - Q = set of irrational numbers.

Question 12:(i) State whether each of the following statement is true or false. Justify your answer.

{ 2, 3, 4, 5 } and { 3, 6} are disjoint sets.

Answer:

Here,

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

{ 2, 3, 4, 5 } $\cap$ { 3, 6} = {3}

Hence,these are not disjoint sets.

So,false.

Question 12:(ii) State whether each of the following statement is true or false. Justify your answer.

{ a, e, i, o, u } and { a, b, c, d }are disjoint sets

Answer:

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

{ a, e, i, o, u } $\cap$ { a, b, c, d } = {a}

Hence,these are not disjoint sets.

So, statement is false.

Question 12:(iii) State whether each of the following statement is true or false. Justify your answer.

{ 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.

Answer:

Here,

{ 2, 6, 10, 14 } and { 3, 7, 11, 15}

{ 2, 6, 10, 14 } $\cap$ { 3, 7, 11, 15} = $\varphi$

Hence, these are disjoint sets.

So,given statement is true.

Question 12:(iv) State whether each of the following statement is true or false. Justify your answer.

{ 2, 6, 10 } and { 3, 7, 11} are disjoint sets.

Answer:

Here,

{ 2, 6, 10 } and { 3, 7, 11}

{ 2, 6, 10 } $\cap$ { 3, 7, 11} = $\varphi$

Hence,these are disjoint sets.

So,statement is true.