JEE Main 2024 — Sets And Relations Question with Solution

To find the number of symmetric relations on the set $$\{1,2,3,4\}$$ that are not reflexive, we first calculate the total number of symmetric relations and then subtract the count of those that are both symmetric and reflexive.

A symmetric relation involves pairs where if a pair (x, y) is in the relation, then (y, x) is also in the relation. For a set with $$n$$ elements, there are $$\frac{n(n+1)}{2}$$ slots in the relation matrix that can independently be occupied or not, corresponding to a total of $$2^{\frac{n(n+1)}{2}}$$ possible symmetric relations.

A relation is reflexive if every element is related to itself, requiring all diagonal slots of the relation matrix (n of them) to be filled. The remaining $$\frac{n(n-1)}{2}$$ slots can be filled in any manner, leading to $$2^{\frac{n(n-1)}{2}}$$ reflexive (and possibly symmetric) relations.

For the set $$\{1,2,3,4\}$$ ($$n=4$$):

• Total symmetric relations: $$2^{\frac{4(4+1)}{2}}=2^{10}=1024$$
• Symmetric and reflexive relations: $$2^{\frac{4(4-1)}{2}}=2^6=64$$

Therefore, the number of symmetric relations that are not reflexive: $$1024-64=960$$.