Guidance on notation for logic and sets

Question

I have been learning more about sets lately and have stumbled upon notation (or I guess a language) such as this:

I would like to know the name of this notation and perhaps some resources to learn it.

Answer 1

The notation is the standard notation of the predicate calculus. But it would be overkill to turn to a textbook on that subject.

$\forall$ means "for all", or "for every". $\varnothing$, as you may know already, is the empty set, i.e., the set with no elements. Likewise $\subseteq$ means "subset of", and $\in$ means "element of".

The slide you've linked to says that the empty set is a subset of any set $B$. It explains that this is true, because the following statement is true:

For every $x$, if $x$ is an element of $\varnothing$, then $x$ is an element of $B$.

The hypothesis, "if $x$ is an element of $\varnothing$", is false, because the empty set has no elements. Any implication whose hypothesis is false is automatically true. (One says, "vacuously true".) You can find many explanations of this fact about implications in logic on the web, including math.stackexchange.