Sets — Case Study Based Questions

Consider the sets given below:

A = {x : x is an integer, $-\dfrac{1}{2}$ < x < $\dfrac{9}{2}$}

B = {x : x is a point in the interior of a circle}

C = {x : x is a perfect square, x < 0}

D = {x : x is a month of the year not having 31 days}

E = {x : x is prime, x < 3}

Look at the given sets and answer the questions given below:

1. Which of the following is an infinite set?
   (a) C
   (b) B
   (c) A
   (d) E

2. Which of the following sets has the minimum number of elements?
   (a) A
   (b) C
   (c) E
   (d) B

3. Which of the following is a singleton set?
   (a) E
   (b) A
   (c) B
   (d) C

4. Which of the following denotes a pair of equivalent sets?
   (a) A and B
   (b) D and A
   (c) A and C
   (d) D and E

Answer

1. Writing each set in roster form:

A = {x : x is an integer, $-\dfrac{1}{2} \lt \text{ x } \lt \dfrac{9}{2}$} = {0, 1, 2, 3, 4}, so n(A) = 5 (finite).

B = {x : x is a point in the interior of a circle} contains an unlimited number of points, so it is an infinite set.

C = {x : x is a perfect square, x < 0} = { }, since no perfect square is negative, so n(C) = 0 (finite).

E = {x : x is prime, x < 3} = {2}, so n(E) = 1 (finite).

Only B is an infinite set.

Hence, option (b) is the correct option.

2. As, n(A) = 5, n(C) = 0, n(E) = 1 and B is infinite.

The set with the minimum number of elements is C, as it is an empty set with n(C) = 0.

Hence, option (b) is the correct option.

3. E = {2} contains exactly one element, so it is a singleton set.

(A has 5 elements, B has infinitely many elements and C is the empty set.)

Hence, option (a) is the correct option.

4. The months of the year not having 31 days are February, April, June, September and November. So, D = {February, April, June, September, November} and n(D) = 5.

Since n(A) = 5, n(C) = 0, n(E) = 1 and B is infinite.

For a pair of equivalent sets, the two sets must have the same cardinal number.

Since n(D) = n(A) = 5, the sets D and A are equivalent.

Hence, option (b) is the correct option.