The set expression $A \cup (B \cap (A' \cup B'))$ is equivalent to

Given expression: $A \cup (B \cap (A' \cup B'))$

Using the distributive law, expand the inner term:

$B \cap (A' \cup B') = (B \cap A') \cup (B \cap B')$

Since $B \cap B' = \phi$, we get:

$B \cap (A' \cup B') = (B \cap A') \cup \phi = B \cap A'$

Substitute this back into the original expression:

$A \cup (B \cap A')$

Apply the distributive law again:

$A \cup (B \cap A') = (A \cup B) \cap (A \cup A')$

Since $A \cup A' = \xi$ (Universal set), we have:

$(A \cup B) \cap \xi = A \cup B$

Answer: $A \cup B$