Mini Review: Sets and Relations

Additional Set Operations

Intersection

The intersection between two sets $\text{X}$ $\text{X}$ and $\text{Y}$ $\text{Y}$ is written as $X\:\cap\:Y$ $X\:\cap\:Y$ and defined as follows:

$X\:\cap\ Y\:=\lbrace x\:|\:x\:\in X\:and\:x\:\in Y\rbrace$ $X\:\cap\ Y\:=\lbrace x\:|\:x\:\in X\:and\:x\:\in Y\rbrace$

Example

For $X=\lbrace a,b\rbrace$ $X=\lbrace a,b\rbrace$ and $Y=\lbrace b,c\rbrace$ $Y=\lbrace b,c\rbrace$ , the intersection is $X\:\cap\:Y=\lbrace b\rbrace$ $X\:\cap\:Y=\lbrace b\rbrace$ .

Difference

The difference between two sets $X$ and $Y$ is written as $X\:\setminus Y$ $X\:\setminus Y$ (or sometimes as X-Y $X-Y$ and is also called the relative complement of $Y$ in $Y$ . It is defined as follows:

$X\:\setminus\:Y=\lbrace x\:|\:x\in X\:and\:x\notin Y\rbrace$ $X\:\setminus\:Y=\lbrace x\:|\:x\in X\:and\:x\notin Y\rbrace$

Example

For $X=\lbrace a,b\rbrace$ $X=\lbrace a,b\rbrace$ and $\lbrace b,c\rbrace$ $\lbrace b,c\rbrace$ , the set difference is $X\:\backslash\:Y=\lbrace a\rbrace$ $X\:\backslash\:Y=\lbrace a\rbrace$ .