To solve the expression $A \cup (A \cup B)'$, we will first break down and analyze the set notation and use a Venn diagram approach to interpret the problem.

Step 1: Understand the components of the expression

1. $A \cup B$ represents the union of sets $A$ and $B$. This is the set of all elements that are in $A$, in $B$, or in both.

2. $(A \cup B)'$ represents the complement of the union of $A$ and $B$. This is the set of all elements that are not in $A \cup B$.

3. $A \cup (A \cup B)'$ represents the union of set $A$ and the complement of $A \cup B$. This is the set of all elements that are either in $A$ or not in $A \cup B$.

Step 2: Set relations using Venn diagram

Let's visualize the sets using a Venn diagram:

• $A \cup B$ would cover all elements in either $A$, $B$, or both.
• The complement of this, $(A \cup B)'$, would include all elements outside of $A \cup B$.
• So, $A \cup (A \cup B)'$ would include all elements in $A$ and all elements outside $A \cup B$.

Step 3: Determine the resulting set

Now, based on the Venn diagram:

• Elements in $A$ are included.
• Elements not in $A \cup B$ are also included (i.e., elements outside both $A$ and $B$).

This means the union $A \cup (A \cup B)'$ includes all elements that are either in $A$ or outside of both $A$ and $B$.

Conclusion:

If we refer to the universal set $U$ as containing all possible elements and assuming we are given the specific values in the universal set and subsets $A$ and $B$, we would find:

$A \cup (A \cup B)' = \{ \text{elements in } A \text{ or outside both } A \text{ and } B \}$

However, since you didn't provide a clear breakdown of the sets $A$, $B$, and the universal set in your question, I can't give a precise list of elements without more information.

Would you like to provide the specific elements in the universal set and sets $A$ and $B$?