The following content is from a book: Rosen discrete mathematics and I understand most of it. But, why can we conclude that $P(a)$ and $Q(a)$ is true for every element in the domain ? Why every element? Because I always have this intuition that "a" is a single element. Is it because "a" is a variable? Can someone explain the reasoning behind?
Content: suppose that $\forall x~(P(x) \wedge Q(x))$ is true. This means that if $a$ is in the domain, then $P(a)\wedge Q(a)$ is true. Hence, $P(a)$ is true and $Q(a)$ is true. Because $P(a)$ is true and $Q(a)$ is true for every element in the domain, we can conclude that $\forall x~P(x)$ and $\forall x~Q(x)$ are both true. This means that $\forall x~P(x)~\wedge~\forall x~Q(x)$ is true.
You can conclude that because $a$ is an arbitrary element of the domain. Because of the universal quantifier in the first statement, $P(a)$ is true for any element $a$ of the domain and we can deduce $\forall x P(x).$
The formal rule is called universal generalization: From $\phi(x),$ conclude $\forall x\phi(x).$