The question is asking for the definition of two sets, $C$ and $D$, being equal.

In set theory, two sets $C$ and $D$ are equal if and only if:

1. Every element of $C$ is an element of $D$ (i.e., $C \subseteq D$).
2. Every element of $D$ is an element of $C$ (i.e., $D \subseteq C$).

This can be summarized as $C = D$ if and only if $C \subseteq D$ and $D \subseteq C$.

Now, reviewing the answer choices:

1. "If every element of $C'$ is also an element of $D$ and there is some element of $D$ that is not an element of $C$" — This does not match the condition for set equality. The complement $C'$ is not relevant here, and the fact that some element of $D$ is not in $C$ directly contradicts equality.

2. "None of the choices shown" — This could be correct if none of the other options meet the conditions for set equality.

3. "If some element of $C$ is also an element of $D$" — This is a weak condition, since just one element being shared doesn't imply equality.

4. "If every element of $C$ is also an element of $D$ and every element of $D$ is also an element of $C$" — This is the correct definition of set equality $C = D$.

5. "If every element of $C$ is also an element of $D$" — This only defines that $C \subseteq D$, but doesn't account for the reverse inclusion $D \subseteq C$, which is also necessary for equality.

Thus, the correct answer is the fourth option: "If every element of $C$ is also an element of $D$ and every element of $D$ is also an element of $C$."

Would you like any further details or clarifications?

Here are 5 related questions:

1. What is the difference between subset and proper subset?
2. How can we verify that two sets are equal using Venn diagrams?
3. What is the role of complements in set theory?
4. How does set equality differ from other set relations like union or intersection?
5. What are common examples of equal sets in mathematics?

Tip: Always remember that set equality requires both inclusions, $C \subseteq D$ and $D \subseteq C$, for a complete proof.