subset and superset

If $A$ and $B$ are sets and every element of $A$ is also an element of $B$, then:

• $A$ is a subset of $B$, written $A\subseteq B$
• $B$ is a superset of $A$, written $B\supseteq A$

If $A\subseteq B$ but $A$ is not equal to $B$ (i.e. there is at least one element of $B$ which is not an element of $A$), then:

• $A$ is a proper (strict) subset of $B$, written $A\subset B$
• $B$ is a proper (strict) superset of $A$, written $A\supset B$

Formally, a subset $S\subseteq X$ is constructed

where the formula ${\mathrm{\Phi }}_S(x)$ is the indicator function of $S$, such that for all $x$ where ${\mathrm{\Phi }}_S(x)$ evaluates to true, $x\in S$, and for all $x$ where $\mathrm{\Phi }(x)$ evaluates to false, $x\notin S$.

The set of all subsets of a set $A$ is called its power set.