Question

Let $A = \{2, 3, 4, 5, \ldots, 16, 17, 18\}$. Let $R$ be the relation on the set $A$ of ordered pairs of positive integers defined by $(a, b) R (c, d)$ if and only if $ad = bc$ for all $(a, b), (c, d) \in A \times A$. Then the number of ordered pairs of the equivalence class of $(3, 2)$ is:

• A. 4
• B. 5
• C. 6
• D. 7

Answer 1

The Correct Option is C

1. Understand the problem:

We have set A = {2, 3, ..., 18} and a relation R defined on A × A such that (a, b)R(c, d) if and only if ad = bc. We need to find the number of ordered pairs in the equivalence class of (3, 2).

2. Find the equivalence class of (3, 2):

All pairs (x, y) must satisfy 3y = 2x ⇒ x/y = 3/2. So we need all pairs in A × A where the ratio x/y = 3/2.

3. List all valid pairs:

Possible pairs (x, y) where both x and y ∈ A and x/y = 3/2:

(3, 2), (6, 4), (9, 6), (12, 8), (15, 10), (18, 12)

4. Count the pairs:

There are 6 such ordered pairs.

Correct Answer: (C) 6