Subset
Elementary
A set $A$ is a subset of a set $B$ if everything in $A$ is also in $B$. For example, the set of golden retrivers is a subset of the set of dogs, and dogs are a subset of all animals.
High school
$A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$.
Properties
- Every set is a subset of itself, and the empty set is a subset of every set.
- $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$.
Notes
The symbol $\subset$ is ambiguous: some authors use it to mean $\subseteq$, and others use it to mean proper subset ($\subsetneq$).