Let $\mathrm{A}=\{1,2,3,4\}$ and R be a relation on the set $A \times A$ defined by $\mathrm{R}=\{((\mathrm{a}, \mathrm{b}),(\mathrm{c}, \mathrm{d})): 2 \mathrm{a}+3 \mathrm{~b}=4 \mathrm{c}+5 \mathrm{~d}\}$. Then the number of elements in R is:

JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Sets Relations and Functions

Let

A = {1, 2, 3, 4}

and R be a relation on the set

A \times A

defined by

R = {((a, b), (c, d)) : 2 a + 3 b = 4 c + 5 d}

. Then the number of elements in R is:

Answer: 6

The relation counts quadruples with

a, b, c, d \in {1, 2, 3, 4}

and

2 a + 3 b = 4 c + 5 d

. The attainable values of

2 a + 3 b

are

{5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20}

, and those of

4 c + 5 d

are

{9, 13, 14, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 31, 32, 36}

, whose common values are

9, 13, 14, 17, 18

. For each common value, multiply the number of

(a, b)

representations by the number of

(c, d)

representations: the value

14

arises from two pairs

(a, b)

but one pair

(c, d)

, contributing

2

, while

9, 13, 17, 18

contribute

1

each. Hence the number of elements in

R

2 + 1 + 1 + 1 + 1 = 6

JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Integer answer