Find count of equivalence relations if you know number of pairs

Question

Set $A$ has seven items. I have to find count of all equivalence relations if $|R|=12$

So if relation is equivalence, than must be reflexive, in $R$ must be seven pairs. So remains $5$ pairs.

I think that count is $0$ because i can't choose $5$ pairs to mantain symmetric and transitivity of relation. Is it right?

Answer 1

Yes it is correct, the number of pairs of the form $(x,y)$ with $x\neq y$ is always even inside any finite equivalence relation. So it cannot be equal to $5$, very nice solution!