Let \(\mathrm{A}=\{1,2,3, \ldots \ldots \ldots 100\}\). Let R be a relation on A defined by \((x, y) \in R\) if and only if \(2 x=3 y\). Let \(\mathrm{R}_{1}\) be a symmetric relation on A such that \(\mathrm{R} \subset \mathrm{R}_{1}\) and the number of elements in \(\mathrm{R}_{1}\) is n. Then, the minimum value of n is ______.

JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Sets Relations and Functions

Let

A = {1, 2, 3, \dots\dots\dots 100}

. Let R be a relation on A defined by

(x, y) \in R

if and only if

2 x = 3 y

. Let

R_{1}

be a symmetric relation on A such that

R \subset R_{1}

and the number of elements in

R_{1}

is n. Then, the minimum value of n is ______.

Answer: 66

R = {(x, y) \in A \times A : 2 x = 3 y}

. For

2 x = 3 y

with

x, y

positive integers,

x

must be a multiple of

3

and

y

a multiple of

2

; write

x = 3 k, y = 2 k

.

For both

x, y \leq 100

3 k \leq 100 \Rightarrow k \leq 33

, and

2 k \leq 100 \Rightarrow k \leq 50

. So

k = 1, 2, \dots, 33

, giving

33

ordered pairs

(3 k, 2 k)

, i.e.

∣ R ∣ = 33

.

For

R_{1}

to be symmetric and contain

R

, it must also contain every reverse pair

(y, x)

for

(x, y) \in R

. None of these reverse pairs

(2 k, 3 k)

coincides with an original pair

(3 k^{'}, 2 k^{'})

R

— equating

3 k^{'} = 2 k

and

2 k^{'} = 3 k

forces

k = k^{'} = 0

, which is excluded — so

R

and its set of reverses are disjoint.

The smallest symmetric relation containing

R

is therefore

R \cup R^{- 1}

, with

n = ∣ R ∣ + ∣ R^{- 1} ∣ = 33 + 33 = 66

JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Integer answer