Relations: Ordered Pairs, Cartesian Product, Relations

If \((x + 1, y - 2) = (3, 1)\), then \((x, y)\) is

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

If \(A = \{a, b, c\}\) and \(B = \{1, 2, 3, 4\}\), then \(n(A \times B)\) is

1. a7
2. b64
3. c12
4. d81

If \(A = \{1, 2\}\) and \(B = \{x, y\}\), then \(A \times B\) is

1. a\(\{(1, x), (1, y), (2, x), (2, y)\}\)
2. b\(\{(x, 1), (y, 1), (x, 2), (y, 2)\}\)
3. c\(\{(1, x), (2, y)\}\)
4. d\(\{(1, 2), (x, y)\}\)

If \(A \times B = \{(p, x), (p, y), (q, x), (q, y), (r, x), (r, y)\}\), then the set \(B\) is

1. a\(\{p, q, r\}\)
2. b\(\{p, x\}\)
3. c\(\{p, q, r, x, y\}\)
4. d\(\{x, y\}\)

Assertion (A): \(A \times \varnothing = \varnothing\) for every set \(A\).

Reason (R): An ordered pair \((a, b)\) in \(A \times \varnothing\) would need \(b \in \varnothing\), which is impossible.

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

If \(n(A) = 3\) and \(n(B) = 2\), the number of relations from \(A\) to \(B\) is

1. a6
2. b8
3. c9
4. d64

1. I. \(\{3, 5\} = \{5, 3\}\)
2. II. \((3, 5) = (5, 3)\)

Which of the statements above is/are correct?

1. aOnly I
2. bOnly II
3. cBoth I and II
4. dNeither I nor II

\(R\) is the relation on \(A = \{1, 2, 3, 4, 5, 6\}\) given by \(R = \{(x, y) : y = x + 2;\ x, y \in A\}\). The range of \(R\) is

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

Match List I with List II:

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

Assertion (A): The empty set \(\varnothing\) is a relation from \(A\) to \(B\).

Reason (R): A relation from \(A\) to \(B\) must contain at least one ordered pair.

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

Let \(A = \{1, 2, 3, 4\}\) and \(B = \{2, 4, 6\}\). \(R\) is the relation from \(A\) to \(B\) in which \((a, b) \in R\) when \(a\) divides \(b\). How many ordered pairs are there in \(R\)?

1. a7
2. b8
3. c6
4. d12

1. I. \(n(A \times B) = n(B \times A)\) for any two finite sets \(A\) and \(B\).
2. II. \(A \times B = B \times A\) for every pair of non-empty sets \(A\) and \(B\).
3. III. If \(A \subseteq B\), then \(A \times A \subseteq B \times B\).

Which of the statements above are correct?

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

\(R = \{(x, y) : x + 2y = 10;\ x, y \in \mathbb{N}\}\), where \(\mathbb{N}\) is the set of natural numbers. The domain of \(R\) is

1. a\(\{1, 2, 3, 4\}\)
2. b\(\{0, 2, 4, 6, 8\}\)
3. c\(\{2, 4, 6, 8, 10\}\)
4. d\(\{2, 4, 6, 8\}\)

Assertion (A): The domain of the relation \(R = \{(1, a), (2, a), (3, b)\}\) is \(\{1, 2, 3\}\).

Reason (R): The range of a relation is always a subset of its codomain.

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

\(A \times B\) has 6 elements, and three of them are \((x, 1)\), \((y, 2)\) and \((z, 1)\), where \(x\), \(y\), \(z\) are distinct. What are the other three elements of \(A \times B\)?

1. a\((x, 2), (y, 2), (z, 2)\)
2. b\((2, x), (1, y), (2, z)\)
3. c\((x, 2), (y, 1), (z, 2)\)
4. d\((1, x), (2, y), (1, z)\)

1. I. The domain of \(R\) is \(A\).
2. II. The range of \(R\) is \(\{2, 4, 6\}\).
3. III. \(R = \{(x, y) : y = 2x;\ x \in A,\ y \in B\}\).

\(R = \{(1, 2), (2, 4), (3, 6)\}\) is a relation from \(A = \{1, 2, 3, 4\}\) to \(B = \{2, 4, 6, 8\}\). Which of the statements above are correct?

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

\(R\) is the relation from \(A = \{1, 2, 3\}\) to \(B = \{1, 4, 9, 16\}\) given by \(R = \{(x, y) : y = x^{2}\}\). Match List I with List II:

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

\(n(A) = 3\), \(n(B) = 4\) and \(n(A \cap B) = 2\). How many elements does \((A \times B) \cap (B \times A)\) have?

1. a2
2. b6
3. c12
4. d4

In a school, S = {Aman, Bela, Chirag} is a set of students and C = {Art, Music} is the set of clubs. The relation R from S to C pairs each student with a club the student has joined: R = {(Aman, Art), (Bela, Art), (Bela, Music)}. The principal wants the set of students who have joined at least one club. Which set gives this?

1. aThe domain of R, {Aman, Bela}
2. bThe range of R, {Art, Music}
3. cThe codomain of R, {Art, Music}
4. dThe set S, {Aman, Bela, Chirag}

A shop records each kind of T-shirt it sells as an ordered pair (size, colour). It sells sizes S, M and L, every size comes in every colour, and there are 15 kinds in all. If the shop adds the size XL in all the same colours, how many kinds will it have?

1. a16
2. b18
3. c20
4. d60