Answer
See step by step work for proof
Work Step by Step
To prove- $A \cup(B\cap C)=(A\cup B)∩(A \cup C)$
Proof-
$A\cup (B\cap C)=${$x|x \in A\cup (B\cap C)$}
Using the definition of union, an element of $A\cup (B\cap C)$ is an element that is in A or in $B\cap C$
={$x|x\in A \lor x\in (B\cap C)$}
Using the definition of intersection, an element of $B\cap C$ is an element that is in B and in C.
={$x|x\in A \lor (x\in B \land x\in C)$}
Using distributive of OR over AND
={$x|(x\in A \lor x\in B) \land (x\in A \lor x\in C)$}
Use the definition of union :
={$x|x\in (A \cup B) \land x\in (A \cup C)$}
Use the definition of intersection :
={$x|x\in ((A \cup B) \cap (A \cup C))$}
=$(A\cup B)∩(A \cup C)$
Thus, we have shown that $A \cup(B\cap C)=(A\cup B)∩(A \cup C)$
