By having the following sentences: $$\alpha=\forall x,y[S(x,y)\rightarrow [S(x,x)\lor y\neq d]]$$ $$\beta=\exists x,y[\lnot S(x,d)\land S(x,y)\land y=d]$$ $$\gamma=\forall x,y[x=d \rightarrow[S(y,y)\lor\lnot S(y,x)]]$$
Which equivalence to which?
I found out that $\alpha$ and $\gamma$ are equivalence since I can use identities and get the exact same thing. We shall use the next identity: $\alpha\rightarrow \beta \equiv \lnot \alpha \lor \beta$. Therefore we have that: $$\alpha = \forall x,y[\lnot S(x,y)\lor S(x,x)\lor y\neq d]$$ and $$\gamma = \forall x,y[x\neq d \lor S(y,y)\lor \lnot S(y,x)]$$ Now, we will exchange $x$ by $y$ and the opposite, which yields: $$\gamma =\forall y,x[\lnot S(x,y) \lor S(x,x)\lor y\neq d] $$ Thus, we have got that $\alpha \equiv \gamma$
However, I can see that $\alpha$ and $\beta$ are not equivalenced because $\beta \equiv \lnot \alpha$, but I am not sure if it is correct. Thank you.