Set Difference

SD#DEF. Set Difference.

The set difference between $X$ and $Y$ is

\[X\setminus Y = \{x\in X:x\notin Y\}.\]

SD#PROP-EMP.

For any $X$:

1. \[X\smallsetminus\varnothing = X.\]
2. \[\varnothing\smallsetminus X = \varnothing.\]
3. \[X\smallsetminus X = \varnothing.\]

SD#PROP-DJ.

If $X$ and $Y$ are disjoint, then

\[\begin{align*} X\smallsetminus Y &= X, \\ Y\smallsetminus X &= Y. \end{align*}\]

SD#PROP-SUB.

If $X\subseteq Y$, then

\[X\smallsetminus Y = \varnothing.\]