The statement $A \subset B \iff A \cap B = A$ is true and can be explained as follows:

Explanation:

• $A \subset B$ means that A is a subset of B, or every element of A is also an element of B.

• $A \cap B$ represents the intersection of sets A and B, meaning it includes only those elements that are in both A and B.

Thus, when $A \subset B$, every element of A is also in B, which implies that:

$A \cap B = A$

because all elements of A are in both A and B, so the intersection would just be A itself. Therefore, the statement $A \subset B \iff A \cap B = A$ holds true.

Recap of the equivalence:

• $A \subset B$ means every element of A is in B.
• $A \cap B = A$ means that the intersection of A and B contains all the elements of A (and possibly others from B).

Thus, both expressions are logically equivalent.

Would you like more details or have any questions?

Related Questions:

1. What is the difference between $A \subseteq B$ and $A \subset B$?
2. Can $A \cap B = A$ if $A$ is not a subset of $B$?
3. How is the intersection $A \cap B$ defined for disjoint sets?
4. What is the relationship between $A \subset B$ and $A \cup B$?
5. What are some examples where $A \cap B = \emptyset$?

Tip:

Remember, when $A \subset B$, this doesn't necessarily mean $A = B$; it simply means all elements of A are contained in B.