Under what condition is the intersection of sets A and B equal to set B?

The intersection of two sets, $A \cap B$, contains all elements that are common to both set A and set B.
If $A \cap B = B$, it means that all elements of set B are also present in set A. If every element of B is also an element of A, then B is a subset of A.

This is written as $B \subseteq A$.
The notation $B \subset A$ typically means B is a proper subset of A (meaning B is a subset of A, and B is not equal to A). However, in multiple-choice questions, $B \subset A$ is often used to represent $B \subseteq A$ when $B \subseteq A$ is not an option.

• A. $A = B^c$: This means A is the complement of B. If A is the complement of B, then A and B have no common elements, so $A \cap B = \emptyset$. This is only equal to B if B is the empty set, which is not generally true.
• B. $A \subset B$: This means A is a subset of B. If A is a subset of B, then $A \cap B = A$. For this to be equal to B, we would need $A = B$. This is not the general condition for $A \cap B = B$.
• C. $A^c = B$: This is the same as option A, just written differently. It implies $A \cap B = \emptyset$, which is not generally true.
• D. $B \subset A$: This means B is a subset of A (or a proper subset). If B is a subset of A, then all elements of B are also in A. Therefore, the elements common to both A and B are precisely the elements of B. So, $A \cap B = B$. This condition is equivalent to $B \subseteq A$.

The condition $A \cap B = B$ is true if and only if $B$ is a subset of $A$.

The final answer is $\boxed{\text{D. } B \subset A}$.