My professor wrote the following theorems and definitions on the blackboard.
Def 1. A ⊂ B ⇔ A ⊆ B ∧ A ≠ B
Thm.1. A ⊂ B → (∀X)(X ∈ B ∧ X ∈ A)
Def.2. (∀X ∈ D)(P(X)) = (∀X)(X ⊆ D → P(X))
(∃X ∈ D)(P(X)) = (∃X)(x ∈ D ∧ P(X))
Thm.2. (∀X ∈ D)(P(X)) is true
(∃X ∈ D)(P(X)) is false
I understand the def 1 and Thm 1 completely, but I have no clue about Def.2 and Thm 2. Can anyone please explain them to me as simple as possible?
Thank you
Definition 2 is defining a shortened form to express domain restrictions.
"Anything in D, makes P true" iff "Anything, when it is in D, then it makes P true".
$$(\forall x\in D)(P(x)) \iff (\forall x)(x\in D\to P(x))$$
"There's something in D, that makes P true" iff "There's something, that is in D and makes P true".
$$(\exists x\in D)(P(x)) \iff (\exists x)(x\in D\wedge P(x))$$
The important point is to learn which connective goes with which quantifier.
For theorem 2, you appear to be missing the declaration that the domain is empty ($D=\emptyset$). When the universe is empty then anything is true and nothing is true.
$(\forall x\in\emptyset)(P(x))$ is vacuously true. It doesn't matter what $P$ is, since there is nothing in the empty set so there's no counter witness.
$(\exists x\in\emptyset)(P(x))$ is false. Again, it doesn't matter what $P$ is as there is nothing in the empty set.
From Definition 2
$$(\forall x\in\emptyset)~(P(x)) \iff \underbrace{(\forall x)\underbrace{(\underbrace{x\in\emptyset}_{\textsf{false}} \to P(x))}_{\textsf{true}}}_{\textsf{true}} \\[2ex] (\exists x\in\emptyset)~(P(x)) \iff \underbrace{(\exists x)~\underbrace{(\underbrace{x\in\emptyset}_{\textsf{false}} \wedge P(x))}_{\textsf{false}}}_{\textsf{false}}$$