Given the sets $A$ and $B$, use the properties of sets to prove that $A \cup (B' \cup A)' = A \cup B$, justifying each step of the proof.

$A \cup (B' \cup A)' = A \cup (B \cap A')$     De Morgan     M1A1

$= (A \cup B) \cap (A \cup A')$     Distributive property     M1A1

$= (A \cup B) \cap U$     (Union of set and its complement)     A1

$= A \cup B$     (Intersection with the universal set)     AG

Note:     Do not accept proofs using Venn diagrams unless the properties are clearly stated.

Note:     Accept double inclusion proofs: M1A1 for each inclusion, final A1 for conclusion of equality of sets.

[5 marks]