Q.Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ.

Concept: Set Difference / Intersection — We need three sets where every pair shares at least one element, but no single element belongs to all three.

Step 1: Let $A = \{1, 2\}$, $B = \{2, 3\}$, $C = \{1, 3\}$.

Step 2: Check pairwise intersections:

$A \cap B = \{2\}$ (non-empty),

$B \cap C = \{3\}$ (non-empty),

$A \cap C = \{1\}$ (non-empty). …

Why this works — the idea of set difference

The condition says: every pair of sets must have some overlap, but the triple intersection must be empty. That means no element can belong to all three sets simultaneously. Yet each pair must share at least one element.

This is a classic Venn diagram puzzle. If you draw three overlapping circles, the triple intersection (the central region where all three overlap) must be empty. But each pair of circles must still touch — meaning each pair's overlap region (outside the centre) must contain at least one element.

The trick is to give each pair its own private shared element, and make sure that element does not appear in the third set.

Step-by-step construction

1. Choose an element for $A \cap B$

Pick $1$ to be in both $A$ and $B$, but not in $C$.

So: $1 \in A$, $1 \in B$, $1 \notin C$.

2. Choose an element for $B \cap C$

Pick $2$ to be in both $B$ and $C$, but not in $A$.

So: $2 \in B$, $2 \in C$, $2 \notin A$.

3. Choose an element for $A \cap C$

Pick $3$ to be in both $A$ and $C$, but not in $B$.

So: $3 \in A$, $3 \in C$, $3 \notin B$.

4. Verify the conditions

• $A \cap B = \{1\}$ — non-empty.
• $B \cap C = \{2\}$ — non-empty.
• $A \cap C = \{3\}$ — non-empty. …

Method: Roster Form Intersection

This method works when sets are given in roster form (listing elements inside curly braces). The intersection of two sets contains only the elements that appear in both sets.

Steps

1. Write both sets clearly in roster form.
2. Identify common elements — scan each element of the first set and check if it is also present in the second set.
3. List only the common elements inside curly braces $\{ \}$.
4. If no element is common, the intersection is the empty set, written as $\emptyset$ or $\{ \}$.

Example (from typical Question 1 type)

Let:

• $A = \{1, 2, 3, 4\}$
• $B = \{3, 4, 5, 6\}$

Step 1: Sets are already in roster form.

Step 2: Common elements:

• $1$ is not in $B$ …

Here are the common mistakes students make with set intersection problems, specifically using the sets you provided, and how to avoid each.

Mistake 1: Forgetting the Definition of "Natural Number" ($\mathbb{N}$)

The Error: Students often assume natural numbers start from 0 or include negative numbers. In the Indian curriculum (NCERT/CBSE), natural numbers are defined as $\{1, 2, 3, 4, ...\}$.

How it affects the answer: If you include 0, then $A \cap B$ might incorrectly include 0. If you include negatives, the intersection with primes ($D$) becomes confusing.

How to Avoid: Memorize the standard definition. For Class 11 NCERT, $\mathbb{N} = \{1, 2, 3, ...\}$. Always write this set down before solving.

Mistake 2: Confusing "Even" and "Odd" with "Prime"

The Error: Students think that because a number is prime, it cannot be even (or odd). They forget that 2 is the only even prime number.

The Consequence: For question (v) $B \cap D$, students often write $\phi$ (empty set) instead of $\{2\}$.

How to Avoid: List the first few elements of each set.

• $B = \{2, 4, 6, 8, 10, ...\}$
• $D = \{2, 3, 5, 7, 11, ...\}$ Now, visually scan for common elements. The only common element is 2.

Mistake 3: Assuming "Odd" and "Prime" are Mutually Exclusive

The Error: Students think that since most primes are odd, the intersection $C \cap D$ must be all odd primes. They forget that 2 is prime but not odd.

The Consequence: For question (vi) $C \cap D$, students write $\{3, 5, 7, 11, ...\}$ (all odd primes) but forget to explicitly exclude 2. While the set of odd primes is correct, the reasoning is flawed if they don't mention that 2 is excluded.

How to Avoid: Always check the boundary case (the number 2).

• $C = \{1, 3, 5, 7, 9, ...\}$
• $D = \{2, 3, 5, 7, 11, ...\}$ The intersection is $\{3, 5, 7, 11, ...\}$ (all odd primes). This is correct, but be explicit: "All prime numbers except 2."

Mistake 4: Writing the Answer in Roster Form Incorrectly

The Error: For infinite sets like $A \cap B$ (which is just $B$), students try to list all elements or write an incomplete roster like $\{2, 4, 6\}$.

The Consequence: Marks are deducted for not showing the pattern or using the wrong notation.

How to Avoid: Use set-builder notation for infinite answers, or use roster form with an ellipsis ($...$).

• Correct: $A \cap B = \{2, 4, 6, 8, ...\}$ or $A \cap B = \{x : x \text{ is an even natural number}\}$.
• Incorrect: $A \cap B = \{2, 4, 6\}$ (this implies the set stops at 6).

Mistake 5: Misinterpreting the Intersection Symbol ($\cap$) …