A={x: x is a two digit number so that sum of its digit is 4} and B={ x: x is a two digit number so that product of its digit is 4) then n(AUB) =?

To find \( n(A \cup B) \), we need to determine the number of elements in the union of sets \( A \) and \( B \). 

First, let's define the sets \( A \) and \( B \):

- \( A = \{ x : x \text{ is a two-digit number such that the sum of its digits is 4} \} \)
- \( B = \{ x : x \text{ is a two-digit number such that the product of its digits is 4} \} \)

### Set \( A \)
We need to find two-digit numbers where the sum of the digits is 4. Let \( x = 10a + b \), where \( a \) and \( b \) are the digits.

\[ a + b = 4 \]

The possible pairs \((a, b)\) that satisfy this equation are:
- \( (1, 3) \)
- \( (2, 2) \)
- \( (3, 1) \)
- \( (4, 0) \)

Therefore, the numbers in set \( A \) are:
- \( 13, 22, 31, 40 \)

So, \( A = \{13, 22, 31, 40\} \).

### Set \( B \)
We need to find two-digit numbers where the product of the digits is 4. Let \( x = 10a + b \), where \( a \) and \( b \) are the digits.

\[ a \times b = 4 \]

The possible pairs \((a, b)\) that satisfy this equation are:
- \( (1, 4) \)
- \( (2, 2) \)
- \( (4, 1) \)

Therefore, the numbers in set \( B \) are:
- \( 14, 22, 41 \)

So, \( B = \{14, 22, 41\} \).

### Union of \( A \) and \( B \)
To find \( A \cup B \), we take the union of sets \( A \) and \( B \):

\[ A \cup B = \{13, 22, 31, 40, 14, 41\} \]

The numbers in \( A \cup B \) are \( 13, 22, 31, 40, 14, 41 \).

Thus, the number of elements in \( A \cup B \) is:

\[ n(A \cup B) = 6 \]

So, \( n(A \cup B) = 6 \).

Would you like details or have any questions about this problem? Here are some related questions for further practice:

1. Find \( n(A \cap B) \) for the sets defined above.
2. Determine \( n(A \setminus B) \).
3. Determine \( n(B \setminus A) \).
4. If \( C = \{ x : x \text{ is a two-digit number such that the difference of its digits is 4} \), find \( n(A \cup C) \).
5. If \( D = \{ x : x \text{ is a two-digit number such that the sum of its digits is 6} \), find \( n(D \cup B) \).
6. Verify if any element in \( A \cap B \) exists.
7. Find the total number of two-digit numbers where the sum of the digits is 5.
8. Find the total number of two-digit numbers where the product of the digits is 6.

**Tip:** When dealing with sets and their unions and intersections, it's helpful to list out the elements clearly to avoid double-counting.

Explore the union of sets A and B, where set A includes two-digit numbers with a digit sum of 4, and set B includes two-digit numbers with a digit product of 4. Understand how to find the number of elements in the union and solve related problems. Suitable for students in Grades 6-8.

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