This page discusses sets, a topic that forms the foundation of arithmetic and real analysis alike.

a collection
a member of a set. We denote $x$ is a element of set $S$ by $x \in S$
empty set, $\varnothing$
the set with no elements
a set $S$ is a subset of $T$, written $S \subset T$, if $ \forall x$ if $ x \in S \Rightarrow x \in T $
finite set
a set in which each element can be matched with exactly one element of the set $\{1, 2, ..., n\}$
set equality
two sets $S$ and $T$ are equal if $S \subset T$ and $T \subset S$ </dl>

Prove: For any set, $S, \varnothing \subset S.$

Consider the statement $x \in \varnothing \Rightarrow x \in S$ which comes from the definition of a subset. $\forall x, x \notin\varnothing.$ This implication's antecedent is always false. Therefore, the implication is always true.