Truth set for the following formula $\forall x R(x, x) \land R(a, b)$?

Question

I have this exercise:

Let $R$ be a binary relation. For each of the following formulas, define a truth set over a universe of size at least 3 satisfying it.

Example: a truth set over a universe of size at least $3$ for the formula $\forall x R(x, x) \land R(a, b)$ is the set {(a, b), (a, a), (b, b), (c, c)} over the universe {a, b, c}.

1. $R(a, b) \land R(b, c)$

2. $\forall x R(x, x)$

3. $\exists x \forall y R(x, y) \land \forall z (z, x)$

I am not sure about my solutions, that's why I am asking.

1. {(a, b), (b, c)}

2. {(a, a), (b, b), (c, c)}

3. {(a, a), (a, b), (a, c), (b, a), (c, a)}

   In this last case, the couples depend of course on the value chosen for $x$, which in my case is $a$. We don't have to repeat $(a, a)$ for both cases.

Are my solutions correct?

Answer 1

Yes, you are correct.

Well, the third is correct if the statement is $\exists x \Big(\forall y R(x, y) \land \forall z R(z, x)\Big)$, and you have chosen $a$ as the existent example of $x$.   (You could have chosen $a$, $b$, $c$, or any combination of at least one of the three).