Hello I've been studying Discrete Mathematics these days but I'm confused that whether existential quantifier include universal quantifier or not.
The meaning of existential quantifier is that "There exists an element x in the domain such that P(x)". I think this quantifier includes all elements in the domain such that P(x) because at least one element exists. So if I think like this, I can conclude that existential quantifier includes universal quantifier.
Is it right or not? I need your help!
The relations between sets, like inclusion, applied to quantifiers are not so intuitive.
According to the standard semantics, to say that $\exists x Px$ is satisfied in a domain $D$ means that there is at least one object in the domain such that $P$ holds of it.
Thus, the subset of the domain $D$ made of all and only those objects $d \in D$ such that $P$ holds of them is not empty.
To say that $\forall x Px$ is satisfied in a domain $D$ means that $P$ holds of every object in the domain.
Thus, the subset of the domain $D$ made of all and only those objects $d \in D$ such that $P$ holds of them is the full domain.
Clearly, "for all" implies "exists", but not vice versa : the fact that there is a cat that is black does not implies that every cat is black.
If we denote with $P^D$ the interpretation of $P$, i.e. $P^D = \{ d \in D \mid P(d) \text { is true in } D \}$ and we denote with $[\![ \varphi ]\!]_D=1$ the fact that the formula $\varphi$ is satisfied in the domain $D$, we have that:
and:
See Generalized quantifiers for the proper representation of quantifiers in terms of sets.