Unions

Definition

The Venn diagram displays two sets $X$ and $Y$ with the union $X\cup Y$ shaded

Let $X$ and $Y$ be subsets of some universal set $U$. The union of $X$ and $Y$, written $X\cup Y$, is the set of all $x$ in $U$ which are in at least one of the sets $X$ or $Y$.
Symbolically, $X\cup Y=\lbrace x\in U:x\in X{\text{ or }}x\in Y\rbrace$.

Examples

Example 1

Determine the union of the sets $X=\lbrace 1,2,5,8\rbrace$ and $Y=\lbrace 1,3,6,9\rbrace$.

Solution By definition, we wish to find the set of all elements which are in at least one of the two sets. Thus, we will collect all unique elements into a set as long as they appear once or more between the sets. Our solution is $X\cup Y=\lbrace 1,2,3,5,6,8,9\rbrace$.

Example 2

Prove that for any sets $X$ and $Y$, $X\subseteq X\cup Y$.

Proof Let $x\in X$. We wish to show that $x\in X\cup Y$, so this means showing that $x\in U$ such that $x\in X$ or $x\in Y$. Since $X\subseteq U$, we have that $x\in U$ and by definition of $x$ we know that $x\in X$. Thus the “or” statement is true and hence $x\in X\cup Y$. This shows that $X\subseteq X\cup Y$.