Extension of a theory

Question

Given is theory $T$ = {$f(x, y) = f(y, x)$} in language only with binary function symbol $f$, (therefore $x, y$ are variables). For each $T_1$ and $T_2$ decide if they are extensions of theory $T$. If yes, show whether they are conservative extensions.

• $T_1$ = {$f(x, x) = x$} in the same language as theory $T$
• $T_2$ = {$∀u∀v(f(u, v) = c$} in the same language but with constant $c$

I'd really appreciate if someone could help me with this as I am completely lost. I know that one way is to show that $M$($T_1$) $\subseteq$ $M$($T$) but how do I show all of the models?

Thank you!

Answer 1

Hints:

• Does every model for $T$ satisfy $f(x, x) = x$?
• Does every model for $T$ satisfy $f(u, v) = f(u', v')$ for all $u, u', v, v'$?