Let `X = {n in N :1 le n le 50}`. If `A = {n in X: " n is a multiple of 2 "}` and `B= {n in X: " n is a multiple of 7"}` , then the number of elements in the largest subset of X containing neither an element of A nor an element of B is_______

Answer

To solve the problem step by step, we will follow these instructions:

1. **Define the Set X**:
We start with the set \( X = \{ n \in \mathbb{N} : 1 \leq n \leq 50 \} \). This set contains all natural numbers from 1 to 50.

**Hint**: Remember that \( \mathbb{N} \) refers to the set of natural numbers.

2. **Identify Set A**:
Set \( A \) consists of elements in \( X \) that are multiples of 2. The multiples of 2 from 1 to 50 are:
\[ A = \{ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50 \} \]
There are 25 elements in set \( A \).

**Hint**: To find multiples of a number, list out the numbers that can be expressed as that number multiplied by integers.

3. **Identify Set B**:
Set \( B \) consists of elements in \( X \) that are multiples of 7. The multiples of 7 from 1 to 50 are:
\[ B = \{ 7, 14, 21, 28, 35, 42, 49 \} \]
There are 7 elements in set \( B \).

**Hint**: Similar to set A, list the multiples of 7 up to 50.

4. **Find the Intersection of A and B**:
We need to find the common elements in sets \( A \) and \( B \):
\[ A \cap B = \{ 14, 28, 42 \} \]
There are 3 elements in the intersection \( A \cap B \).

**Hint**: The intersection includes elements that are common to both sets.

5. **Calculate the Union of A and B**:
We can use the formula for the union of two sets:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Substituting the values we found:
\[ |A \cup B| = 25 + 7 - 3 = 29 \]

**Hint**: The union of two sets combines all unique elements from both sets.

6. **Find the Complement of A Union B in X**:
We want to find the largest subset of \( X \) that contains neither elements of \( A \) nor elements of \( B \). This is the complement of \( A \cup B \) in \( X \):
\[ |(A \cup B)'| = |X| - |A \cup B| = 50 - 29 = 21 \]

**Hint**: The complement of a set includes all elements not in that set.

7. **Conclusion**:
The number of elements in the largest subset of \( X \) containing neither an element of \( A \) nor an element of \( B \) is \( 21 \).

**Final Answer**: 21