I'm totally new to predicate logic and I need to make an interpretation that will be true for both formulas:
$$ \varphi = \forall x \forall y (\forall z(E(z,x)\Leftrightarrow E(z,y))\Rightarrow x=y) $$ $$ \psi = \forall z \exists w \forall y \forall x ((E(x,y) \wedge E(y,z)) \Rightarrow E(x,w)) $$
According to my assignment, i have such domain: $$ U=\{a,b\} $$ $$ |[E]| = \{(a,a),(a,b)\} $$
My idea was set predicate symbol E this way.
$$ E(x,y)\:\:=\:\:x !=y $$
Did i do it right?