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Proof

(related to Proposition: Distributivity Laws For Sets)

• By hypothesis, $A,B,C$ are sets.
• We will prove that the distributivity laws hold for the set union "$\cup$" and the set intersection "$\cap$".

Ad "$(1)$"

• The first law follows from the distributivity laws for conjunction and disjunction and the respective definitions of the set union and the set intersection:

$$\begin{array}{rcl} x\in A\cap (B\cup C) &\Leftrightarrow&x\in A\wedge x\in (B\cup C)\\ &\Leftrightarrow&x\in A\wedge (x\in B\vee x\in C)\\ &\Leftrightarrow&(x\in A\wedge x\in B)\vee (x\in A\wedge x\in C)\\ &\Leftrightarrow&x\in (A\cap B)\vee x\in (A\cap C)\\ &\Leftrightarrow&x\in (A\cap B)\cup (A\cap C) \end{array}$$

Ad "$(2)$"

• Again, we use the distributivity laws for conjunction and disjunction and the respective definitions of the set union and the set intersection:

$$\begin{array}{rcl} x\in A\cup (B\cap C) &\Leftrightarrow&x\in A\vee x\in (B\cap C)\\ &\Leftrightarrow&x\in A\vee (x\in B\wedge x\in C)\\ &\Leftrightarrow&(x\in A\vee x\in B)\wedge (x\in A\vee x\in C)\\ &\Leftrightarrow&x\in (A\cup B)\wedge x\in (A\cup C)\\ &\Leftrightarrow&x\in (A\cup B)\cap (A\cup C) \end{array}$$

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References

Bibliography

1. Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016