The following table represent the truth values of a predicate Q(X,Y), where X in the rows and Y in the columns. I need to evaluate the truth status (T, F) of the attached statements. I am not sure how to approach this question. Can you please guide me? My intuition say, for example, that if I have "for all X, Y exist such that Q(X,Y), then I need to check if for every row, there is at least one T. Am I correct? How do I solve 8 then ? Thank you.
Question $8$ says:
$$ \forall y Q(y,y) \land \exists x \forall y Q(x,y) $$
We can break this question up into the left and right condition which must both be satisfied for this statement to be true.
This is just question 3. It says, for any assignment to $y$ from the domain, $Q(y,y)$ holds. In other words, everything in the domain is true with itself. This is true since the diagonal of the truth table is all T's.
This is just question 6. It says there is some value that is true with everything else i.e. you can show me that there is some row in the truth table which is all T's. The second row ($x=b$) is one such example. Thus this is true as well.
Since both conjuncts are true, the original statement is true.