Q.Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = { 2, 3, 5, 7 }. Find A ∩ B and hence show that A ∩ B = B

Concept: Set Intersection and Set Equality

The intersection $A \cap B$ consists of all elements that belong to both $A$ and $B$ simultaneously.

Given $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $B = \{2, 3, 5, 7\}$.

Scanning through $B$: every element $2, 3, 5, 7$ is present in $A$. Therefore,

$A \cap B = \{2, 3, 5, 7\}$ …

Why intersection captures "common membership"

The intersection of two sets is the collection of elements that satisfy a double requirement: membership in the first and membership in the second. Think of it as the overlap in a Venn diagram. When we write $A \cap B$, we are asking: which elements pass the test for both $A$ and $B$?

In this problem, $A$ is the set of the first ten natural numbers, and $B$ is the set of prime numbers less than $10$. Notice that $B$ is already "sitting inside" $A$ — every prime in $B$ is automatically one of the numbers from $1$ to $10$. This observation will make the intersection straightforward.

Finding $A \cap B$ step by step

1. List the elements of $B$:

   $B = \{2, 3, 5, 7\}$.

2. Check each element of $B$ for membership in $A$:

   • Is $2 \in A$? Yes, $2$ is in $\{1, 2, 3, \ldots, 10\}$.
   • Is $3 \in A$? Yes.
   • Is $5 \in A$? Yes.
   • Is $7 \in A$? Yes.

3. Collect the common elements:

   Every element of $B$ is also in $A$, so

4. Compare $A \cap B$ with $B$: …

1. Concept First — The Idea of Set Intersection

The intersection of two sets is one of the most fundamental ideas in set theory. When we say A ∩ B (read as "A intersection B"), we are asking:

"Which elements are common to both sets A and B?"

Think of it like a Venn diagram — two overlapping circles. The intersection is the overlapping region where both circles contain the same items.

Why does this question ask us to "show that A ∩ B = B"?

This is a special case. When every element of B is also in A, then the intersection of A and B is simply B itself. In other words:

If B is a subset of A (written B ⊆ A), then A ∩ B = B.

Here, B = {2, 3, 5, 7} — these are all prime numbers less than 10. And A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} contains all numbers from 1 to 10. So clearly, every element of B lives inside A. That's the intuition.

2. Step-by-Step Solution

Step 1: List the elements of A and B clearly

We have:

• A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
• B = {2, 3, 5, 7}

Step 2: Find the common elements (the intersection)

We go through each element of B and check if it is present in A:

Since all four elements of B are also in A, the intersection contains exactly these four elements.

So:

Step 3: Compare with B

We see that:

They are identical sets. Therefore:

3. Final Answer

4. Why It Works & Exam Tip

Why it works …

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 ($\cap$) …