The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is _________.

<p>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.</p><p>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.</p><p>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.</p><p>For the set $\{1,2,3,4\}$ ($n=4$):</p><ul><li>Total symmetric relations: $2^{\frac{4(4+1)}{2}} = 2^{10} = 1024$</li><li>Symmetric and reflexive relations: $2^{\frac{4(4-1)}{2}} = 2^{6} = 64$</li></ul><p>Therefore, the number of symmetric relations that are not reflexive: $1024 - 64 = 960$.</p>