Q.Find the intersection of each pair of sets of question 1 above.

Intersection means the elements common to both sets. Using the five pairs of sets from Question 1 of this exercise:

1. $\{1,3,5\}\cap\{1,2,3\}=\{1,3\}$
2. $\{a,e,i,o,u\}\cap\{a,b,c\}=\{a\}$
3. (multiples of 3) $\cap$ (naturals less than 6) $=\{3\}$ …

This question reuses the five pairs of sets from Question 1 of the same exercise and asks for the intersection ($\cap$) of each pair -- the elements common to both sets.

(i) $X=\{1,3,5\}$, $Y=\{1,2,3\}$. The elements appearing in both are $1$ and $3$.

(ii) $A=\{a,e,i,o,u\}$, $B=\{a,b,c\}$. Only $a$ appears in both.

(iii) $A = \{x : x \text{ is a natural number and a multiple of } 3\} = \{3,6,9,12,\dots\}$, $B = \{x : x \text{ is a natural number less than } 6\} = \{1,2,3,4,5\}$. The only multiple of 3 that is also less than 6 is $3$.

(iv) $A = \{x : x \text{ is a natural number}, 1 < x \le 6\} = \{2,3,4,5,6\}$, $B = \{x : x \text{ is a natural number}, 6 < x < 10\} = \{7,8,9\}$. No number can be both $\le 6$ and $> 6$ at once.

$A \cap B = \emptyset$ …

Method: Direct Element Comparison (Set Intersection)

Concept: The intersection of two sets $A$ and $B$, written $A \cap B$, is the set of all elements that belong to both $A$ and $B$.

Steps

1. List all elements of the first set clearly.
2. List all elements of the second set clearly.
3. Compare each element of the first set with the elements of the second set.
4. Pick only those elements that appear in both sets.
5. Write the result as a set (curly braces, elements separated by commas).

Example (from a typical Question 1)

Let’s assume the pairs from Question 1 were:

• Pair (i): $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$
• Pair (ii): $C = \{a, b, c\}$, $D = \{b, c, d, e\}$

For Pair (i):

• Elements of $A$: $1, 2, 3, 4$
• Elements of $B$: $3, 4, 5, 6$
• Common elements: $3$ and $4$
• Answer: $A \cap B = \{3, 4\}$

For Pair (ii): …

Here’s a breakdown of the common mistakes students make when finding the intersection of sets (based on a typical "Question 1" style from NCERT or state board textbooks), along with how to avoid each.

✗ Mistake 1: Confusing Intersection with Union

What students do:

They list all elements from both sets instead of only the common ones.

Example:

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

Wrong answer: $A \cap B = \{1, 2, 3, 4\}$ (this is actually $A \cup B$).

Why it happens:

Students mix up the symbols $\cap$ (intersection) and $\cup$ (union).

✓ How to avoid:

• Memorise the shape: $\cap$ looks like an arch — think “Arch for And” (both sets must have it).
• Always ask: “Is this element in both sets?” If no, leave it out.

✗ Mistake 2: Forgetting to Check All Elements in Both Sets

What students do:

They only check elements from the first set against the second, but miss elements that appear only in the second set.

Example:

$X = \{a, b, c\}$, $Y = \{b, c, d\}$

Wrong: $X \cap Y = \{b\}$ (missed $c$).

Why it happens:

Rushing — they stop after scanning the first set.

✓ How to avoid:

• Systematic method:
  1. List all elements of set 1.
  2. For each, check if it’s in set 2.
  3. Then repeat by checking elements of set 2 against set 1 (or just verify your list).
• Use a Venn diagram to visualise overlap.

✗ Mistake 3: Including Duplicate Elements in the Answer

What students do:

They write the same element twice in the intersection set.

Example:

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

Wrong: $A \cap B = \{2, 2, 3\}$

Why it happens:

They copy duplicates from the original sets without realising sets don’t have repetitions.

✓ How to avoid:

• Remember: Sets never contain duplicates.
• Write the intersection as a set of distinct elements only.
• After listing, check: “Did I write any number twice?”

✗ Mistake 4: Writing the Answer in Wrong Notation

What students do:

They write the answer as a list without curly braces, or use round brackets.

Example:

Wrong: $A \cap B = 2, 3$ or $(2, 3)$

Why it happens:

Carelessness with set notation.

✓ How to avoid:

• Always use curly braces $\{ \}$ for sets.
• Double-check your final answer format before moving to the next question.

✗ Mistake 5: Assuming Intersection Always Has Elements …