Discrete Math: Set Theory

Set Theory


Let $x$ be any element. $x$ is an element of $A \cup (B \cap C)$ if and only if $x$ is an element of $A$ or $x$ is an element of $B \cap C$. This is true if and only if $x$ is an element of $A$ or ($x$ is an element of $B$ and $x$ is an element of $C$). This is true if and only if ($x$ is an element of $A$ or $x$ is an element of $B$) and ($x$ is an element of $A$ or $x$ is an element of $C$). This is true if and only if $x$ is an element of $(A \cup B) \cap (A \cup C)$. Therefore, $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

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