The minimum number of elements that must be added to the relation $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c}),(\mathrm{b}, \mathrm{d})\}$ on the set $\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$ so that it is an equivalence relation, is ____.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Sets Relations and Functions

The minimum number of elements that must be added to the relation

R = {(a, b), (b, c), (b, d)}

on the set

{a, b, c, d}

so that it is an equivalence relation, is ____.

Answer: 13

Solution

An equivalence relation must be reflexive, symmetric and transitive. The pairs

(a, b), (b, c), (b, d)

tie

a

b

and

b

c

and

d

, so symmetry and transitivity pull all four elements

a, b, c, d

into a single equivalence class. The only equivalence relation containing the given pairs therefore relates every element to every element, consisting of all

4 \times 4 = 16

ordered pairs. Three of these are already present, so the number of pairs that must be added is

16 - 3 = 13

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Integer answer

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