Hello! Merry Christmas and a Happy New Year.

I got one problem.

Alpha is relation on A x A, where A is set.

I need to prove:

alpha is symmetric and antisymmetric if and only if alpha is subset of {(x,x)|x is in A}

I tried to prove it this way.

Let x,y is in the set A.

Alpha is symetric:

If (x,y) is in alpha then (y,x) is in alpha.

Alpha is antisymetric:

If (x,y) is in alpha and (y,x) is in alpha then x=y

So I need to prove:

(P -> Q) ^ ((P ^ Q) -> S) <=> (P -> S)

where p denotes - (x,y) is in alpha

q denotes - (y,x) is in alpha

s denotes - x=y

But when I construct truth table (it is not tautology) it falls on:

P- true

Q-false

S-true

Is the theorem not correct or I did some mistake?

Thanks in advance.