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

Answer 2

The relation \( R \) is defined as \( (a, b)R(c, d) \) if and only if \( ad = bc \). This is equivalent to saying that the fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) are equal. In other words, \( R \) defines an equivalence relation where \( (a, b) \) and \( (c, d) \) are equivalent if they represent the same rational number.

The equivalence class of \( (3, 2) \) consists of all ordered pairs \( (a, b) \) such that \( \frac{a}{b} = \frac{3}{2} \). Since \( A = \{2, 3, \dots, 18\} \), we need to find pairs \( (a, b) \) in \( A \times A \) that satisfy this ratio.

We can express this as \( \frac{a}{b} = \frac{3}{2} \), which means \( a = \frac{3}{2}b \). We need to find integer values of \( b \) such that \( \frac{3}{2}b \) is also an integer and both \( a \) and \( b \) are in \( A \).

Let's try some values:

• If \( b = 2 \), \( a = 3 \). \( (3, 2) \) is already in the equivalence class.
• If \( b = 4 \), \( a = 6 \). \( (6, 4) \)
• If \( b = 6 \), \( a = 9 \). \( (9, 6) \)
• If \( b = 8 \), \( a = 12 \). \( (12, 8) \)
• If \( b = 10 \), \( a = 15 \). \( (15, 10) \)
• If \( b = 12 \), \( a = 18 \). \( (18, 12) \)
• If \( b = 14 \), \( a = 21 \), which is not in \( A \).

Thus, we have the following ordered pairs in the equivalence class of \( (3, 2) \): \( (3, 2), (6, 4), (9, 6), (12, 8), (15, 10), (18, 12) \).

There are 6 ordered pairs in the equivalence class of \( (3, 2) \).