Sets: Representation, Subsets, Intervals, Venn Diagrams

Which of the following collections is a set?

1. aThe collection of the most talented students in a class
2. bThe collection of the vowels of the English alphabet
3. cThe collection of the beautiful cities of India
4. dThe collection of the good books in a library

The set of all prime numbers less than 12, written in roster form, is

1. a\(\{1, 2, 3, 5, 7, 11\}\)
2. b\(\{3, 5, 7, 11\}\)
3. c\(\{2, 3, 5, 7, 11\}\)
4. d\(\{2, 3, 5, 7, 9, 11\}\)

Which of the following is the set-builder form of \(\{1, 4, 9, 16, 25\}\)?

1. a\(\{x : x = n^{2}, n \in \mathbb{N}, n \le 5\}\)
2. b\(\{x : x = n^{2}, n \in \mathbb{N}, n \le 25\}\)
3. c\(\{x : x = 2n - 1, n \in \mathbb{N}, n \le 5\}\)
4. d\(\{x : x \in \mathbb{N}, x \le 25\}\)

If \(A = \{a, b, c, d\}\), how many elements does the power set \(P(A)\) have?

1. a8
2. b4
3. c15
4. d16

The power set of a set \(A\) has 64 elements. How many elements does \(A\) have?

1. a32
2. b6
3. c8
4. d64

In interval notation, \(\{x : x \in \mathbb{R}, -2 < x \le 5\}\) is written as

1. a\([-2, 5)\)
2. b\((-2, 5)\)
3. c\((-2, 5]\)
4. d\([-2, 5]\)

Assertion (A): The empty set \(\phi\) is a subset of every set \(A\).

Reason (R): \(\phi\) has no elements, so there is no element of \(\phi\) that is not in \(A\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): \(\{1, 2, 3\}\) and \(\{a, b, c\}\) are equivalent sets.

Reason (R): Any two sets that have the same number of elements are equal sets.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): \(\{0\}\) is an empty set.

Reason (R): An empty set has no elements.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. \(\{x : x \in \mathbb{N}, 2 < x < 3\}\) is an empty set.
2. II. \(\{x : x \in \mathbb{N}, x + 3 = 5\}\) is a singleton set.
3. III. The set of all natural numbers that are multiples of 5 is a finite set.

Which of the statements about types of sets is/are correct?

1. aOnly I and II
2. bOnly I
3. cOnly II and III
4. dI, II and III

1. I. Every set is a subset of itself.
2. II. If \(A = \{a, b\}\), then \(a \in P(A)\).
3. III. A set with 3 elements has 7 proper subsets.

Which of the statements about subsets and power sets is/are correct?

1. aOnly I
2. bOnly I and III
3. cOnly II and III
4. dI, II and III

1. I. \(7 \in (3, 7)\)
2. II. The interval \((2, 5)\) contains exactly two integers.
3. III. \([-1, 4) \subset [-1, 4]\)

Which of the statements about intervals is/are correct?

1. aOnly I
2. bOnly I and II
3. cI, II and III
4. dOnly II and III

Let \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), \(A = \{1, 2, 3, 4, 5\}\) and \(B = \{4, 5, 6, 7\}\). Match List I with List II:

1. a(i)-S, (ii)-P, (iii)-R, (iv)-Q
2. b(i)-Q, (ii)-R, (iii)-P, (iv)-S
3. c(i)-S, (ii)-R, (iii)-P, (iv)-Q
4. d(i)-S, (ii)-R, (iii)-Q, (iv)-P

For two sets \(A\) and \(B\) in a universal set \(U\), match each set in List I with the part of their Venn diagram described in List II:

1. a(i)-P, (ii)-R, (iii)-S, (iv)-Q
2. b(i)-R, (ii)-P, (iii)-S, (iv)-Q
3. c(i)-R, (ii)-P, (iii)-Q, (iv)-S
4. d(i)-R, (ii)-S, (iii)-P, (iv)-Q

If \(n(A) = 25\), \(n(B) = 18\) and \(n(A \cup B) = 35\), then \(n(A \cap B)\) is

1. a8
2. b10
3. c17
4. d43

In a class of 40 students, 25 like cricket, 20 like football and 10 like both games. How many students like neither cricket nor football?

1. a15
2. b0
3. c10
4. d5

A shop surveyed 60 customers: 38 bought tea, 30 bought coffee and 6 bought neither. How many customers bought exactly one of the two drinks?

1. a14
2. b40
3. c54
4. d24

Let \(A = \{1, 3, 5\}\), \(B = \{2, 4, 6\}\) and \(C = \{0, 2, 4, 6, 8\}\). Which of the following can be taken as a universal set for all three sets?

1. a\(\{1, 2, 3, 4, 5, 6, 7, 8\}\)
2. b\(\{0, 2, 4, 6, 8\}\)
3. c\(\{0, 1, 2, 3, 4, 5, 6, 7, 8\}\)
4. d\(\{0, 1, 2, 3, 4, 5, 6\}\)

Which of the following pairs of sets are equal?

1. aThe letters of FOLLOW and the letters of WOLF
2. b\(\{x : x^{2} = 4, x \in \mathbb{Z}\}\) and \(\{2\}\)
3. c\(\{x : x \in \mathbb{N}, x < 4\}\) and \(\{0, 1, 2, 3\}\)
4. d\(\{1, 2, 3\}\) and \(\{a, b, c\}\)

If \(A = (-3, 5]\) and \(B = [2, 8)\) are intervals of real numbers, then \(A - B\) is

1. a\([2, 5]\)
2. b\((-3, 2]\)
3. c\((5, 8)\)
4. d\((-3, 2)\)