Subsets

Definition: Subset

A set $A$ is a subset of the set $B$ if all elements of $A$ are also elements of $B$.

$$a\in A⟹a\in B$$

NOTATION

If $A$ is a subset of $B$, we write $A\subseteq B$.

If $A$ is not a subset of $B$, we write $A⊈B$.

Definition: True Subset

A set $A$ is a true subset of the set $B$ if $A\subseteq B$ and $A\ne B$.

$$(a\in A⟹a\in B)\wedge (\exists b\in B:b\notin A)$$

In other words, there are elements in $B$ which are not elements of $A$.

NOTATION

If $A$ is a true subset of $B$, we write $A\subset B$.

If $A$ is a subset, but not a true subset, of $B$, we write $A⊊B$.

NOTE

True subset are also known as strict subsets.