Let's first break down the sets:

• Set $A = \{ x \mid x \text{ is an even integer} \}$, which includes all even integers like $\ldots, -4, -2, 0, 2, 4, 6, 8, \ldots$.
• Set $B = \{ 4, 5, 6, 7 \}$.

Now, we need to find the union $A \cup B$, which means combining all elements from both sets without duplicating any.

• From $A$, we take all even integers.
• From $B$, we take $\{4, 5, 6, 7\}$.

So, the union will be the set of all even integers combined with $5$ and $7$ (since 4 and 6 are already included in the even integers):

$A \cup B = \{ x \mid x \text{ is an even integer or } x = 5 \text{ or } x = 7 \}$

Thus, the correct answer is: A. $\{ x \mid x \text{ is an even integer or } x = 5 \text{ or } x = 7 \}$

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Would you like more details or have any questions?

Here are five related questions for further practice:

1. What is the intersection of sets $A$ and $B$?
2. If $C = \{2, 3, 6\}$, what is $A \cup C$?
3. Can the union of two sets be smaller than either set?
4. How does the union of infinite sets differ from finite sets?
5. What is the symmetric difference between $A$ and $B$?

Tip: When finding the union of two sets, always include all distinct elements from both sets.