JEE Mathematics 2023 Question with Solution

Given: $R = \{(a, b), (b, c), (b, d)\}$ on the set $\{a, b, c, d\}$.

Find: The minimum number of ordered pairs that must be added so that $R$ becomes an equivalence relation.

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

For reflexivity, the pairs

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

must be present.

For symmetry, since

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

we must also add

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

so that each ordered pair has its reverse.

For transitivity, the required additional pairs are

$$(a,c),(a,d),(c,a),(d,a),(c,d),(d,c)$$

.

Thus, the total number of pairs to be added is

$$4 + 3 + 6 = 13$$

.

Therefore, the minimum number of elements to be added is $13$.

Given: $R = \{(a, b), (b, c), (b, d)\}$ on $\{a, b, c, d\}$.

Find: How many more ordered pairs are needed to make it an equivalence relation.

Because $a$ is related to $b$, $b$ is related to $c$, and $b$ is related to $d$, all four elements must lie in the same equivalence class after completion. Hence the final equivalence relation on $\{a,b,c,d\}$ must contain all possible ordered pairs of this class, namely $4^2 = 16$ pairs.

The relation already contains $3$ pairs:

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

.

So the number of new pairs required is

$$16 - 3 = 13$$

.

Therefore, the minimum number of elements to be added is $13$.