Q.Show that A ∩ B = A ∩ C need not imply B = C.

The key idea is that set difference (or a counterexample using disjoint sets) shows the implication fails — the intersection with $A$ only constrains elements inside $A$, leaving elements outside $A$ free to differ.

Reasoning:

1. We need sets $A, B, C$ such that $A \cap B = A \cap C$ but $B \neq C$.
2. Choose $A = \{1\}$, $B = \{1, 2\}$, $C = \{1, 3\}$.
3. Then $A \cap B = \{1\}$ and $A \cap C = \{1\}$, so the condition holds. …

The question asks us to show that $A \cap B = A \cap C$ does not force $B = C$. This is a classic lesson about the limits of set intersection.

Think about what $A \cap B$ captures: it only "sees" the part of $B$ that lies inside $A$. Anything in $B$ that is outside $A$ is invisible to the intersection. So if two sets $B$ and $C$ differ only in elements that are not in $A$, their intersections with $A$ will be identical. The intersection is blind to differences outside $A$.

To prove that a statement is not always true, we only need one counterexample — a single concrete case where the premise holds but the conclusion fails.

1. Choose a small universal set to keep things clear. Let's work with $U = \{1, 2, 3, 4\}$.

2. Pick a set $A$ that will be our "window". Let $A = \{1, 2\}$.

3. Construct $B$ and $C$ so that they share the same elements inside $A$ but differ outside $A$.

   Let $B = \{1, 2, 3\}$ and $C = \{1, 2, 4\}$.

4. Compute the intersections:

   $A \cap B = \{1, 2\} \cap \{1, 2, 3\} = \{1, 2\}$

   $A \cap C = \{1, 2\} \cap \{1, 2, 4\} = \{1, 2\}$

   So $A \cap B = A \cap C$ holds.

5. Check whether $B = C$:

   $B = \{1, 2, 3\}$ and $C = \{1, 2, 4\}$. …

Method: Counterexample by Set Construction

Method Name: Counterexample Method (using specific finite sets)

Why This Works

The statement "if $A \cap B = A \cap C$, then $B = C$" is not always true. To prove this, we need just one example where the premise holds but the conclusion fails. This is the essence of disproving a universal claim.

Steps

Step 1: Choose a simple universal set

Pick a small universe, say $U = \{1, 2, 3, 4\}$.

Step 2: Construct set $A$ to be non-empty but not too large

Let $A = \{1, 2\}$.

Step 3: Construct $B$ and $C$ such that:

• Their intersections with $A$ are identical
• But $B$ and $C$ themselves are different

Let:

• $B = \{1, 3\}$
• $C = \{1, 4\}$

Step 4: Verify the premise

Compute $A \cap B$:

• $A \cap B = \{1, 2\} \cap \{1, 3\} = \{1\}$

Compute $A \cap C$:

• $A \cap C = \{1, 2\} \cap \{1, 4\} = \{1\}$

So $A \cap B = A \cap C = \{1\}$ ✓

Step 5: Check the conclusion

• $B = \{1, 3\}$
• $C = \{1, 4\}$

Clearly, $B \neq C$ because $3 \in B$ but $3 \notin C$, and $4 \in C$ but $4 \notin B$. ✓

Final Result …

Here are the common mistakes students make when tackling this problem, along with clear strategies to avoid them.

Mistake 1: Trying to prove the statement is always false

• The Error: Students often try to prove that $A \cap B = A \cap C$ must imply $B = C$. They attempt algebraic manipulation (e.g., using distributive laws) to force a proof, getting stuck or making a logical leap.
• Why it happens: The question asks to "show that it need not imply" — meaning you must find a counterexample. The statement is not a theorem; it's a caution that the implication is false in general.
• How to avoid: Immediately recognize that the task is to disprove the implication. Your goal is to find one specific set $A$, $B$, and $C$ where $A \cap B = A \cap C$ but $B \neq C$.

Mistake 2: Choosing sets where $A$ is empty

• The Error: A student writes $A = \emptyset$, $B = \{1\}$, $C = \{2\}$. Then $A \cap B = \emptyset = A \cap C$, and $B \neq C$. This is technically correct, but it's a trivial counterexample.
• Why it happens: It's the easiest way to make the intersection equal — an empty intersection always works.
• How to avoid: While not wrong, examiners often expect a non-trivial example where $A$ is non-empty. A better example: Let $A = \{1, 2\}$, $B = \{1, 3\}$, $C = \{1, 4\}$. Then $A \cap B = \{1\} = A \cap C$, but $B \neq C$. This shows the concept more clearly.

Mistake 3: Forgetting to explicitly state $B \neq C$

• The Error: A student writes $A = \{1\}$, $B = \{1, 2\}$, $C = \{1, 3\}$, computes $A \cap B = \{1\} = A \cap C$, and stops. They never explicitly say "Therefore $B \neq C$."
• Why it happens: They assume the reader will infer it, but in exams, explicit conclusion is required.
• How to avoid: After showing $A \cap B = A \cap C$, always write: "But $B \neq C$ because $2 \in B$ but $2 \notin C$ (or vice versa)."

Mistake 4: Using sets that are not well-defined or ambiguous

• The Error: Writing something like $A = \text{all even numbers}$, $B = \text{all multiples of 4}$, $C = \text{all multiples of 2}$ — without specifying the universal set or being precise.
• Why it happens: Students try to be clever with infinite sets but lose clarity.
• How to avoid: Stick to finite, explicitly listed sets for counterexamples. For example: $A = \{1, 2, 3\}$, $B = \{1, 2, 4\}$, $C = \{1, 2, 5\}$. Then $A \cap B = \{1, 2\} = A \cap C$, and $B \neq C$ (since $4 \in B$ but $4 \notin C$).

Mistake 5: Confusing intersection with union or difference …