Q.Determine whether the following statement is true or false. Justify your answer: For all sets $A$, $B$ and $C$, if $A \subset B$, then $A \cap C \subset B \cap C$.

Concept: Set Difference / Subset relation under intersection.

We need to check whether $A \subset B$ forces $A \cap C \subset B \cap C$.

Take any element $x \in A \cap C$.

• By definition of intersection, $x \in A$ and $x \in C$.
• Since $A \subset B$, $x \in A$ implies $x \in B$.
• So $x \in B$ and $x \in C$, hence $x \in B \cap C$. …

This is a classic exercise in understanding what the subset symbol $\subset$ really means, and how intersection interacts with it. Many students get confused because they think of "subset" as a kind of containment that might break when you intersect with another set — but it doesn't.

Let’s unpack why.

1. Restate what we need to prove

We are given: $A \subset B$. That means every element of $A$ is also an element of $B$. We want to check whether $A \cap C \subset B \cap C$ must follow.

In words: if we take the elements that are in both $A$ and $C$, are they necessarily also in both $B$ and $C$?

2. Pick an arbitrary element from $A \cap C$

Let $x \in A \cap C$. By definition of intersection, this means:

• $x \in A$
• $x \in C$

Since $x \in A$ and $A \subset B$, we know $x \in B$.

So now we have:

• $x \in B$ (from the subset condition)
• $x \in C$ (from the intersection)

3. Conclude that $x$ belongs to $B \cap C$

By definition of intersection again, $x \in B$ and $x \in C$ together mean $x \in B \cap C$.

We started with an arbitrary $x \in A \cap C$ and showed $x \in B \cap C$. That is exactly the definition of $A \cap C \subset B \cap C$.

4. A common misunderstanding …