ThmDex – An index of mathematical definitions, results, and conjectures.

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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

▶	Set complement
▶	Symmetric set difference

Results

▶	Antitonicity of subtracting from the same set
▶	Difference of set and binary union equals intersection of differences
▶	Difference of set and countable union equals intersection of differences
▶	Difference of set and finite union equals intersection of differences
▶	Isotonicity of subtracting same set
▶	Set difference equals intersection with complement
▶	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 $X \setminus Y$.