If A = { x : x = `3n-1, n in N and n le 5 `}
B = { x : x is an odd natural number and `x lt 15 `} and
C = { x : x =4n , ` n in N and n lt 7` } ,find
`A cap B `

Answer

To solve the problem, we need to find the intersection of sets A and B. Let's break down the steps:

### Step 1: Define Set A
Set A is defined as:
\[ A = \{ x : x = 3n - 1, n \in \mathbb{N} \text{ and } n \leq 5 \} \]

We will calculate the elements of A by substituting values of \( n \) from 1 to 5.

- For \( n = 1 \):
\[ x = 3(1) - 1 = 2 \]
- For \( n = 2 \):
\[ x = 3(2) - 1 = 5 \]
- For \( n = 3 \):
\[ x = 3(3) - 1 = 8 \]
- For \( n = 4 \):
\[ x = 3(4) - 1 = 11 \]
- For \( n = 5 \):
\[ x = 3(5) - 1 = 14 \]

Thus, the elements of set A are:
\[ A = \{ 2, 5, 8, 11, 14 \} \]

### Step 2: Define Set B
Set B is defined as:
\[ B = \{ x : x \text{ is an odd natural number and } x < 15 \} \]

The odd natural numbers less than 15 are:
\[ B = \{ 1, 3, 5, 7, 9, 11, 13 \} \]

### Step 3: Find the Intersection of Sets A and B
The intersection \( A \cap B \) consists of elements that are common to both sets A and B.

From the elements of A:
\[ A = \{ 2, 5, 8, 11, 14 \} \]

And the elements of B:
\[ B = \{ 1, 3, 5, 7, 9, 11, 13 \} \]

Now, we will identify the common elements:
- The number 5 is in both A and B.
- The number 11 is also in both A and B.

Thus, the intersection \( A \cap B \) is:
\[ A \cap B = \{ 5, 11 \} \]

### Final Answer
The intersection of sets A and B is:
\[ A \cap B = \{ 5, 11 \} \]

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