Definition 1. The set difference between two sets $X$ and $Y$ is
\[X \setdif Y = \{x \in X : x \notin Y\}.\]Proposition 1. For any set $X$:
- \[X \setdif \empt = X.\]
- \[\empt \setdif X = \empt.\]
- \[X \setdif X = \empt.\]
Proposition 2. Any two sets $X$ and $Y$ are disjoint if and only if
\[X \setdif Y = X.\]Proposition 3. For any two sets $X$ and $Y$, if $X \subseteq Y$, then
\[X \setdif Y = \empt.\]References
- [1]
- Wikipedia contributors. Complement (set theory). Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 8.