Answer
See step by step answer for proof.
Work Step by Step
To proof- $ A \cup (B \cup C)=(A \cup B) \cup C$
Proof-$ A \cup (B \cup C)$={$x|x \in A \cup (B \cup C)$}
Using the definition of union, an element of $ A \cup (B \cup C)$ is an element of $A$ or an element of $(B \cup C)$
={$x|x \in A \lor x \in (B \cup C)$}
Using the definition of union, an element of $(B \cup C)$ is an element that is in $B$ or in $C$.
={$x|x \in A \lor (x \in B \lor x \in C)$}
Using the associative law:
={$x|(x \in A \lor x \in B) \lor x \in C$}
Using the definition of union:
={$x|x \in (A \cup B) \lor (x \in C)$}
Use the definition of union again:
={$x|x \in (A \cup B) \cup C$}
=$(A \cup B) \cup C$
Thus we have shown that $A \cup (B \cup C)$=$(A \cup B) \cup C$
