RBSE Class 11 Maths Important Questions Chapter 1 Sets

Rajasthan Board RBSE Class 11 Maths Important Questions Chapter 1 Sets Important Questions and Answers.

RBSE Class 11 Maths Chapter 1 Important Questions Sets

Question 1.
{B = y: y is a vowel of English alphabet}, its Roster form :
(A) {a, e, i, o}
(B) {a, o, u}
(C) {a, e, o, u}
(D) {a, e, i, o, u)
Answer:
Vowels of English alphabet are a, e, i, o, u. Thus, the set of vowels is {a, e, i, o, u}.
Hence, option (D) is correct.

Question 2.
Set-builder form of set A = {1, 4, 9, 16, 25, } will be :
(A) {x : x is an odd natural number}
(B) {x : x is an even natural number}
(C) {x : x is square of natural number}
(D) {x : x is a prime natural number}
Answer:
Elements of set A = {1, 4, 9, 16, 25, .........} are the square of 1, 2, 3,4, 5, ....... respectives.
Hence, option (C) is correct.

Question 3.
Which set is infinite set of the following :
(A) {x : x ∈ Vand (x - l)(x - 2) = 0}
(B) {x : x ∈ N and x2 = 4}
(C) {x : x ∈ N and 2x - 1 = 0}
(D) {x : x ∈ N and x is a prime number}
Answer:
Option (D) is correct, because element of this set is a prime number which is infinite. Hence, this set is infinite.

Question 4.
If A = {0}, B = {x: x > 15 and x < 5}, C = {x : x - 5 = 0}, D = {x: x² = 25}, E = {x : x is a positive integer root of equation x² - 2x - 15 = 0}, then pair of equal sets is :
(A) A, B
(B) B, C
(C) C, D
(D) C, E
Answer:
Option (D) is correct, as in the tabular form the set C = {5} and set E = {5}
So, C = E
∴ Pair of equal sets is C and E.

Question 5.
Which of the following statement is incorrect:
(A) if A and B are dis joint set then A ∩ B = Φ
(B) If (A ∪ B) = Φ then A and B both empty set always.
(C) If (A ∩ B) = Φ then A = Φ and B is any set
(D) If (A ∩ B) = A then B is empty set ‘
Answer:
Option (D) is incorrect.

Question 6.
If A ∩ B = Φ, then :
(A) A - B = Φ
(B) A - B = A
(C) A ∪ B = Φ
(D) A - B = B
Answer:
∵ A ∩ B = Φ thus, Φ, is common in A and B both. So, other elements of A is not present in B.
Thus A - B = A
Hence, option (B) is correct.

Question 7.
Shaded portion of the following Venn diagram represents:

(A) A ∪ B
(B) A ∩ B
(C) A - B
(D) B - A
Answer:
Shaded portion shows that all the elements in A which is not in B. i.e., A B.
Hence, option (C) is correct.

Question 8.
If A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}, then A - B will be :
(A) {1, 3, 5, 8}
(B) {1/3, 5}
(C) {1, 2, 3, 4, 5, 6, 8)
(D) = { }
Answer:
Given, A = {T, 2, 3,4, 5, 6}
and B = {2, 4, 6, 8}
A - B means A contains the element which is not present in B. .
Thus, A - B = {1, 3, 5} .
Hence, option (B) is correct.

Question 9.
If U = {a, b, c, .... x, y, z}, then write three sets A, B, C where A ∩ B ≠ Φ, B ∩ C ≠ Φ, C ∩ A ≠ Φ but A ∩ B ∩ C ≠ Φ.
Answer:
Let A = {a. b, c, d, e, f}
B = {d, e, f, i, j, k}
C = {a, b, c, i, j, k}
A ∩ B = {d, e, f} ≠ Φ
B ∩ C = {i, J, k} ≠ Φ
C ∩ A = {a, b, c) ≠ Φ
(A ∩ B) ∩ C = A ∩ (B ∩ C)
{d, e, f} ∩ {a, b, c, i, j, k} = {a, b, c, d, e, f} ∩ {i, j, k} = Φ

Question 10.
Prove that
(A - B) ∪ (B - A) = (A ∪ B) - (A ∩ B)
Answer:
x ∈ (A - B) ∪ (B - A) ⇒ x ∈ (A - B) or x ∈ (B - A)
⇔ (x ∈ A and x ∉ B) or (x ∈ B and x ∉ A)
⇔ {x ∈ A and x ∉ B or x ∉ B} and
{x ∈ A or x ∈ B and x ∉ A}
⇔ {x ∈ B or x ∈ A and x ∉ B}
and {x ∈ B or (x ∈ A and x ∉ B)}
⇔ {(x ∈ B or x ∈ A) and (x ∈ B or x ∈ B)} and {(x ∉ A or x ∈ B)}
⇔ {x ∈ A ∪ B and x ∈ U} and {x ∈ U and x ∉ (A ∩ B)}
⇔ x ∈ {(A ∩ B) and x ∉ (A ∩ B)}
⇔ x ∈ {(A ∪ B) - (A ∩ B)}
Hence Proved.

Question 11.
Prove that
A ∩ (B - C) = (A ∩ B) - (A ∩ C)
Answer:
Let x ∈ A ∩ (B - C) ⇒ x ∈ A and x ∈(B - C)
⇒ x ∈ A and (x ∈ B and x ∉ C)
⇒ (x ∈ A and x ∈ B) and (x ∈ A and x ∉ C)
= x ∈ (A ∩ B and x ∈ (A ∩ C)
⇒ x ∈(A ∩ B) - (A ∩ C)
A ∩ (B - C) ⊆ (A ∩ B) - (A ∩ C) .............. (1)
Thus, Let
x ∈ {(A ∩ B) - (A ∩ C)}
⇒ x ∈ (A ∩ B) and x ∈ (A ∩ C)
⇒ x ∈ (A ∩ B) and x ∈ C
⇒ x ∈ A and (x ∈ B and x ∈ C)
⇒ x ∈ A and x ∈ (B - C)
⇒ x ∈ A ∩ (B - C)
(A ∩ B - (A ∩ C) ⊆ A ∩ (B - C) ................ (2)
Now, from equation (1) and (2)
A ∩ (B - C) = (A ∩ B) - (A ∩ C)
Hence Proved.

Question 12.
If S = {(x, : |x - 3| < 1 and | y - 31 < 1}
E = {(x, y): 4x² + 9y² - 32x - 54y + 109 ≤ 0}, then show that S ⊂ E.
Answer:
|x - 3| < 1 ⇒ - 1 < x - 3 ⇒ 2 < x < 4
|y - 3|< 1 ⇒ - 1 < y - 3 ⇒ 2 < y < 4
Thus S is set of those points lies between the square formed by x = 2, x = 4, y = 2, y = 4
4x² + 9y² - 32x - 54y + 109 ≤ 0
4(x² - 8x + 16) + 9(y² - 6y + 9) - 36 < 0
4(x - 4)² + 9(y - 3)² - 36 ≤ 0
\frac{(x-4)^2}{9}+\frac{(y-3)^2}{4} ≤ 1
Thus, E is set of those points which lies at given ellipse and inside it.

From figure, it is clear that S ⊂ E.
Hence Proved.

Multiple Choice Questions

Question 1.
If A and B are two sets such that A ⊂ B, then A ∪ B is:
(a) A
(b) B
(c) Φ
(d) None of these
Answer:
(b) B

Question 2.
If A and B are two sets, then A - B equals :
(a) {x : x ∈ A and x ∉ B}
(b) {x : x ∉ A and x ∉ B)
(c) {x : x ∉ A and x ∈ B
(d) None of these
Answer:
(a) {x : x ∈ A and x ∉ B}

Question 3.
If A' is compliment of A, then (A')' equals :
(a) A
(b) A'
(c) Φ
(d) None of these
Answer:
(a) A

Question 4.
Symmetric difference of two sets A and B is :
(a) (A - B) ∪ (B - A)
(b) (A - B) ∩ (B - A)
(c) Φ
(d) None of these
Answer:
(a) (A - B) ∪ (B - A)

Question 5.
If set A, B are such that n(A) = 20, n(B) = 25 and n(A ∪ B) = 40, then n(A ∩ B) is :
(a) 20
(b) 40
(c) 5
(d) None of these
Answer:
(c) 5

Question 6.
If A and B are disjoint sets, then n(AuB) equals:
(a) n(A) + n(B) - n(A ∩ B)
(b) n(A) + n(B)
(c) n(A) + n(B) + n(A ∩ B)
(d) None of these
Answer:
(b) n(A) + n(B)

Question 7.
If A = {2, 4, 6, 8, 10}, then proper subsets of set A are:
(a) 5
(b) 31
(c) 30
(d) 32
Answer:
(b) 31

Question 8.
If two sets A and B are such that A ⊂ B, then which of the following statement is true ?
(a) B' ⊆ A'
(b )B' ⊄ A'
(c) A' ⊆ B'
(d) None of these
Answer:
(a) B' ⊆ A'

Question 9.
If A, B and C are three sets, then A ∪ (B ∩ C) equals:
(a) (A ∪ B) ∪ (A ∪ C)
(b) (A ∪ B) ∩ (A ∪ C)
(c) (A ∩ B) ∪ (A ∩ C)
(d) None of these
Answer:
(b) (A ∪ B) ∩ (A ∪ C)

Question 10.
If A, B and C are the three sets, then A ∩ (B ∪ C) equals:
(a) (A ∩ B) ∪ (A ∩ C)
(b) (A ∩ B) ∩ (A ∪ C)
(c) (A ∪ B) ∩ (A ∪ C)
(d) None of these
Answer:
(a) (A ∩ B) ∪ (A ∩ C)

Question 11.
If A = {x : x ∈ R and x > 8}, B = {x : x ∈ R and x ≤ 9}, then A ∩ B equals:
(a) {8, 9}
(b) {8}
(c) {9}
(d) None of these
Answer:
(c) {9}

Question 12.
If A and B are sets, then (A - B) ∪ (B - A) equals :
(a) (A - B) ∩ B
(b) (B - A) ∪ A
(c) (A ∪ B) - (A ∩ B)
(d) None of these
Answer:
(c) (A ∪ B) - (A ∩ B)

Question 13.
If A and B are sets, then A - B equals :
(a) B' - A'
(b) A ∩ B
(c) A' ∩ B
(d) None of these
Answer:
(a) B' - A'

Question 14.
For set A and B, A - (A - B) equals: —
(a) A ∩ B
(b) A ∪ B
(c) Φ
(d) None of these
Answer:
(a) A ∩ B

Question 15.
If U is universal set containing 800 elements and its subsets A and B are such that n(A) = 250, n(B) = 350, n(A ∩ B) = 150, then n(A ' ∩ B')equals:
(a) 250
(b) 350
(c) 450
(d) None of these
Answer:
(b) 350

Question 16.
If A and B are sets then n(A' ∩ B') equals :
(a) n(A ∪ B)'
(b) n(A ∪ B)
(c) n(A ∩ B)'
(d) None of these
Answer:
(a) n(A ∪ B)'

Question 17.
For sets A and B, A ∪ B = A if and only if:
(a) A ⊆ B
(b) B ⊆ A
(c) A = B
(d) A ≠ B
Answer:
(b) B ⊆ A

Question 18.
If for sets A and B, A ∩ B = A, then :
(a) A ⊂ B
(b) B ⊂ A
(c) A = B
(d) None of these
(a) A ⊂ B

Fill in the blanks in each of the questions:

Question 1.
If A = {a, b, c, d, e},B = {d, e}, then A - B is equal to ..............................
Answer:
{a, b, c}

Question 2.
Power set of the set A = {3, 5} is ................................
Answer:
{Φ, {3}, {5}, {3, 5}}

Question 3.
No. of improper subset of any set having n elements are ................................
Answer:
2ⁿ - 1

Question 4.
If A and B are disjoint sets then A ∩ B is ..................................
Answer:
Φ

Question 5.
If A and B are finite sets such that A ⊂ B, then n (A ∪ B) - .............................
Answer:
n(B)

Question 6.
If A and B are two sets, then A ∩ (A ∪ B) equals to ............................
Answer:
A

Question 7.
If n (A) = 5 then number of subset of A is .........................
Answer:
32

Question 8.
If A and B are disjoint set then A - B is equal .............................
Answer:
A

Question 9.
If any set has only one element them it is called .............................
Answer:
singlten set.

State true or false for the following statements in each questions:

Question 1.
For any two sets A ∩ B = Φ always.
Answer:
False

Question 2.
If A and B are disjoint sets then
n(A ∪ B) = n (A) + n (B)
Answer:
True

Question 3.
If X = {1, 2, 3} and Y = {x:x ∈ R and 1 ≤ x ≤ 3}, then X = Y.
Answer:
False

Question 4.
Set {a, b, c, d} and {d, c, b, a} are equal sets.
Answer:
True

Question 5.
If A = {Φ, 1, 2, {3}, {4}, 5}, then (1, 2, {3}} ∈ A.
Answer:
False

Question 6.
For any two set A and B if P (A) = P (B), then A = B,
Answer:
True

Question 7.
For any two sets n (A ∩ B) > n (A).
Answer:
False

Question 8.
The smallest set A such that A ∪ {1, 2} = {1, 2, 3, 4} is {2, 3, 4, 5}.
Answer:
False

Question 9.
The set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, .......} is finite set.
Answer:
False