ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Definition D70
Set difference
Formulation 0
Let $X$ and $Y$ each be a D11: Set.
The difference of $X$ and $Y$ is the D11: Set \begin{equation} X \setminus Y : = \{ x \in X : x \not\in Y \} \end{equation}
Formulation 1
Let $X$ and $Y$ each be a D11: Set.
The difference of $X$ and $Y$ is the D11: Set \begin{equation} X \setminus Y : = \{ x : x \in X \text{ and } x \not\in Y \} \end{equation}
Children
D79: Set complement
D71: Symmetric set difference
Results
R2062: Antitonicity of subtracting from the same set
R4173: Difference of set and binary union equals intersection of differences
R4171: Difference of set and countable union equals intersection of differences
R4172: Difference of set and finite union equals intersection of differences
R49: Isotonicity of subtracting same set
R50: Set difference equals intersection with complement
R47: Set difference is subset of the minuend
Conventions
Convention 0 (Notation for set difference)
If $X$ and $Y$ are each a D11: Set, we denote the D70: Set difference of $X$ and $Y$ by $X \setminus Y$.