Definition – Subset. – Math-reference.net

Definition

Let $A$ and $B$ be sets, and let $x\in{A}$ . We say that $A$ is a subset of $B$ if $x\in{A}\implies{x\in{B}}$ . We may also say that $A$ is contained in $B$ .

If $x\in{A}\implies{x\in{B}}$ , we write that $A\subseteq{B}$ , and if $A\neq{B}$ , we say that $A$ is a proper subset of $B$ , and denote this by $A\subset{B}$ .

Notes

We can see from the definition that proving $A$ is a (proper) subset of $B$ is equivalent to proving $x\in{A}\implies{x\in{B}}$ .

The empty set

The empty set $\emptyset$ is a subset of every set. The proof is given here.