Thinking Mathematically (6th Edition)

$A\cup (A\cup B)'=\{23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71\}$
In the diagram $(A\cup B)$ is a set of all elements inside the circle for $A$, the circle for $B$, or in both the circle for $A$ and the circle for $B$. $(A\cup B)'$ is a set of elements in $U$ (the whole diagram) but not in $(A\cup B)$. We can see that those elements are: $(A\cup B)' = \{53, 59, 61, 67, 71\}$ The elements in $A$ from the diagram are: $A=\{23, 29, 31, 37, 41, 43\}$ Therefore: $A\cup (A\cup B)'=\{23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71\}$