Subset, Superset and Proper Subset

A set $A$ is called a subset of a set $B$ — written $A \subseteq B$ — if every element of $A$ is also an element of $B$. Formally,

When $A \subseteq B$, we also say $B$ is a superset of $A$, written $B \supseteq A$.

Two useful facts follow immediately: every set is a subset of itself ($A \subseteq A$, since every element of $A$ is trivially in $A$), and the empty set is a subset of every set ($\emptyset \subseteq A$ for any $A$, since it has no elements that could ever fail to be in $A$).

A set $A$ is a proper subset of $B$ — written $A \subset B$ — if $A \subseteq B$ and $A \ne B$; that is, every element of $A$ is in $B$, but $B$ contains at least one element that $A$ does not. A worked example the syllabus itself points to: let $\mathbb{N} = \{1, 2, 3, \dots\}$ and $\mathbb{W} = \{0, 1, 2, 3, \dots\}$. Every natural number is a whole number, so $\mathbb{N} \subseteq \mathbb{W}$; but $0 \in \mathbb{W}$ while $0 \notin \mathbb{N}$, so $\mathbb{N} \ne \mathbb{W}$, which makes $\mathbb{N} \subset \mathbb{W}$ — a proper subset.