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

Result R50 on D70: Set difference

Set difference equals intersection with complement

Formulation 0

Let $X, A, B$ each be a D11: Set such that

(i)	$A, B \subseteq X$ are each a D78: Subset of $X$

Then \begin{equation} B \setminus A = B \cap (X \setminus A) \end{equation}

Proofs

Proof 0

Let $X, A, B$ each be a D11: Set such that

(i)	$A, B \subseteq X$ are each a D78: Subset of $X$

Proceeding directly from the definitions, we have \begin{equation} \begin{split} B \setminus A & = \{ x : x \in B \text{ and } x \not\in A \} \\ & = \{ x : x \in B \subseteq X \text{ and } x \not\in A \} \\ & = \{ x : x \in B \text{ and } x \in X \text{ and } x \not\in A \} \\ & = \{ x : x \in B \text{ and } x \in X \setminus A \} \\ & = B \cap (X \setminus A) \end{split} \end{equation} $\square$