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

Question

Accepted Answer

> [!TLDR]
> The intersection $A \cap B$ collects all elements that belong to **both** sets. Since every element of $B$ already lies in $A$, we get $A \cap B = \{2, 3, 5, 7\} = B$.

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

4. **Compare $A \cap B$ with $B$:** …

Operation	What it asks	Bucket contains
Union ( $A \cup B$ )	In either?	Everything from $A$ or $B$ (or both)
Intersection ( $A \cap B$ )	In both?	Only the overlap
Difference ( $A \setminus B$ )	In $A$ but not $B$ ?	Just the part of $A$ that doesn't touch $B$
Complement ( $A^c$ )	Not in $A$ ?	Everything outside $A$ (inside $U$ )
Symmetric Difference ( $A \triangle B$ )	In exactly one?	The two crescent-shaped parts, excluding the overlap

Element of B	Is it in A?	Reason
2	✓ Yes	2 is between 1 and 10
3	✓ Yes	3 is between 1 and 10
5	✓ Yes	5 is between 1 and 10
7	✓ Yes	7 is between 1 and 10

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 understanding — Set Operations

The idea in plain words

Why this works

Step by step

A common slip …

Why intersection captures "common membership"

Finding $A \cap B$ step by step

1. Concept First — The Idea of Set Intersection

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

2. Step-by-Step Solution

Step 1: List the elements of A and B clearly

Step 2: Find the common elements (the intersection)

Step 3: Compare with B

3. Final Answer

4. Why It Works & Exam Tip

Why it works …

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

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

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

Mistake 4: Writing the Answer in Roster Form Incorrectly

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

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

Why intersection captures "common membership"

Finding A∩BA \cap BA∩B step by step

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Finding $A \cap B$ step by step

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