Subsets

Definition

Let $X$ and $Y$ be sets. We say that $X$ is a subset of $Y$ if every element of $X$ is also an element of $Y$ , and we write $X\subseteq Y$ or $Y\supseteq X$ . Symbolically, $X\subseteq Y$ means $x\in X$ $\Rightarrow$ $x\in Y$ .

Two sets $X$ and $Y$ are said to be equal, $X=Y$, if both $X\subseteq Y$ and $Y\subseteq X$. Note that some authors use the symbol $\subset$ in place of the symbol $\subseteq$.