Distributive laws for sets

Statement

Let $A$, $B$ and $C$ be sets. Then intersection distributes over union, and union distributes over intersection: $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C),$$ $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C).$$

Remarks

The laws follow from the distributive laws of logic for $\land$ and $\lor$. The second law has no analogue for numbers: $a + bc$ is not $(a + b)(a + c)$ in general.

Explanations

L1 human

To prove the first law, take $x$ in the left side. Then $x \in A$, and $x \in B$ or $x \in C$. In the first case $x \in A \cap B$, and in the second case $x \in A \cap C$. The other inclusion is similar.