Q:

how to give a truth value for the following formula

I am trying give a structure that makes that makes the formula T and a structure that makes the formula F for the following formula
∀X•(p(X) ⇔ q(X)) ∧ ∃Y•(p(Y) ∨ q(X))

so I thought for the T value I could use the domain of all humans say that p is Love and q is happy. So the meaning would be if X is in love means the X is happy and vice versa but I am not sure what do for Y. 

A:

The formula has a free occurrence of the variable symbol $X$. For clarity, we replace the bound occurrences of $X$ by $Z$, so we are looking at the formula 
$$\forall z(p(Z)\iff q(Z))\land \exists Y(p(Y)\lor q(X)).\tag{1}$$
Some would not assign a truth value to formulas with free occurrences of variable symbols. Some would assign to such a formula the same truth value as to the sentence obtained by universally quantifying all free occurrences of variable symbols. We take the latter point of view.
To make the formula (1) true, we can use the following interpretation. Let the universe of discourse consist of a single object $a$, and let $p(a)$ and $q(a)$ be true. 
To make the formula (1) false, we can let the universe consist of a single object $a$, and let $p(a)$ and $q(a)$ be false. The $\forall Z(p(Z)\iff q(Z))$ part of (1) is true, but the $\exists Y(p(Y)\lor q(X))$ part is false, so their conjunction (1) is false.

