If A is a subset of B (A B), which of the following statements is false?

17. C. $A \cap B = B$

Given that $A \subset B$, this means that every element of set $A$ is also an element of set $B$.

Let's evaluate each statement:

• A. $A \cap B = A$ Since all elements of $A$ are in $B$, the elements common to both $A$ and $B$ are precisely the elements of $A$. This statement is TRUE.

• B. $A \cup B = B$ Since all elements of $A$ are already in $B$, combining $A$ and $B$ results in the set $B$ itself. This statement is TRUE.

• C. $A \cap B = B$ This statement implies that the intersection of $A$ and $B$ is equal to $B$. For this to be true, $B$ must be a subset of $A$ ($B \subset A$). However, we are given $A \subset B$. If both $A \subset B$ and $B \subset A$ were true, then $A$ and $B$ would be equal ($A=B$). But $A \subset B$ does not necessarily mean $A=B$. For example, if $A = \{1\}$ and $B = \{1, 2\}$, then $A \subset B$. In this case, $A \cap B = \{1\}$, which is $A$, not $B$. Therefore, this statement is generally FALSE.

• D. Every element of A is in B This is the definition of $A \subset B$. This statement is TRUE.

The final answer is $\boxed{\text{C}}$.