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
`B cap C `

Answer

To solve the problem, we need to find the intersection of sets B and C. 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 need to calculate the values for \( n = 1, 2, 3, 4, 5 \):
- 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,
\[ A = \{ 2, 5, 8, 11, 14 \} \]

### 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: Define Set C
Set C is defined as:
\[ C = \{ x : x = 4n, n \in \mathbb{N} \text{ and } n < 7 \} \]
We need to calculate the values for \( n = 1, 2, 3, 4, 5, 6 \):
- For \( n = 1 \): \( x = 4(1) = 4 \)
- For \( n = 2 \): \( x = 4(2) = 8 \)
- For \( n = 3 \): \( x = 4(3) = 12 \)
- For \( n = 4 \): \( x = 4(4) = 16 \) (not included since \( n < 7 \))
- For \( n = 5 \): \( x = 4(5) = 20 \) (not included since \( n < 7 \))
- For \( n = 6 \): \( x = 4(6) = 24 \) (not included since \( n < 7 \))

Thus,
\[ C = \{ 4, 8, 12 \} \]

### Step 4: Find the Intersection of Sets B and C
Now we need to find \( B \cap C \):
- \( B = \{ 1, 3, 5, 7, 9, 11, 13 \} \)
- \( C = \{ 4, 8, 12 \} \)

The intersection \( B \cap C \) consists of elements that are common to both sets. Since all elements in B are odd and all elements in C are even, there are no common elements.

Thus,
\[ B \cap C = \emptyset \]

### Final Answer
The intersection of sets B and C is:
\[ B \cap C = \emptyset \]

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