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]