QJEEM24Ma925d: JEE Main 2024 Mathematics - Class 11 JEEM24Ma925d Solution: The number of symmetric relations defined on the set which are…

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: $1024 - 64 = 960$ .