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

Result R2223 on D77: Set union

Binary set union is commutative

Formulation 0

Let $X$ and $Y$ each be a D11: Set.

Then \begin{equation} X \cup Y = Y \cup X \end{equation}

Proofs

Proof 0

Let $X$ and $Y$ each be a D11: Set.

Proceeding directly from the definitions \begin{equation} \begin{split} X \cup Y & = \{ x : x \in X \text{ or } x \in Y \} \\ & = \{ x : x \in Y \text{ or } x \in X \} \\ & = Y \cup X \end{split} \end{equation} $\square$

Proof 1

Let $X$ and $Y$ each be a D11: Set.

This result is a particular case of R4214: Finite set union is invariant under bijective shifting of indices. $\square$