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Question

# Prove the first associative law by showing that if A, B, and C are sets, then A ∪ (B ∪ C) = (A ∪ B) ∪ C.

Solution

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DEFINITIONS

$\textbf{Union}$ $A\cup B$: All elements that are either in $A$ OR in $B$

$\textbf{Intersection}$ $A\cap B$: All elements that are both in $A$ AND in $B$.

$\textbf{Complement }\overline{A}$: All elements in the universal set $U$ NOT in $A$.

$\textbf{Negation}$ $\neg p$: not $p$

$\textbf{Disjunction}$ $p\vee q$: $p$ or $q$

$\textbf{Conjunction}$ $p\wedge q$: $p$ and $q$

$\emptyset$ represents the $\textbf{empty set}$ and the empty set does not contain any elements (nor sets).

$\textbf{Associative laws for propositions}$:

$(p \vee q) \vee r\equiv p\vee (q\vee r)$

$(p \wedge q) \wedge r\equiv p\wedge (q\wedge r)$

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