Let $\mathrm{A}=\{1,2,3, \ldots 20\}$. Let $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ two relation on A such that $\mathrm{R}_{1}=\{(\mathrm{a}, \mathrm{b}): \mathrm{b}$ is divisible by a$\}$ $\mathrm{R}_{2}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}$ is an integral multiple of b$\}$. Then, number of elements in $\mathrm{R}_{1}-\mathrm{R}_{2}$ is equal to ______.

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Sets Relations and Functions

Let

A = {1, 2, 3, \dots 20}

. Let

R_{1}

and

R_{2}

two relation on A such that

R_{1} = {(a, b) : b

is divisible by a

}

R_{2} = {(a, b) : a

is an integral multiple of b

}

.
Then, number of elements in

R_{1} - R_{2}

is equal to ______.

Answer: 46

A = {1, 2, \dots, 20}

R_{1} = {(a, b) : a ∣ b}

;

R_{2} = {(a, b) : b ∣ a}

. Note

(a, b) \in R_{2} ⟺ (b, a) \in R_{1}

n (R_{1}) = a = 1 \sum 20 ⌊ \frac{20}{a} ⌋ = 20 + 10 + 6 + 5 + 4 + 3 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 66

R_{1} \cap R_{2}

requires

a ∣ b

and

b ∣ a

simultaneously, forcing

a = b

: these are the 20 pairs

(1, 1), \dots, (20, 20)

n (R_{1} - R_{2}) = n (R_{1}) - n (R_{1} \cap R_{2}) = 66 - 20 = 46

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Integer answer