If `A={x,y,z}` and `B={1,2}`, then the number of relations from A to B is

Options

A. 32
B. 16
C. 64
D. 128

Answer

To find the number of relations from set A to set B, we can follow these steps:

### Step 1: Identify the sets
We have two sets:
- \( A = \{x, y, z\} \)
- \( B = \{1, 2\} \)

### Step 2: Determine the Cartesian product \( A \times B \)
The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set A and the second element is from set B. We can list these pairs:
- \( (x, 1) \)
- \( (x, 2) \)
- \( (y, 1) \)
- \( (y, 2) \)
- \( (z, 1) \)
- \( (z, 2) \)

Thus, the Cartesian product \( A \times B \) is:
\[ A \times B = \{(x, 1), (x, 2), (y, 1), (y, 2), (z, 1), (z, 2)\} \]

### Step 3: Count the number of elements in \( A \times B \)
From the list, we can see that there are 6 elements in \( A \times B \):
1. \( (x, 1) \)
2. \( (x, 2) \)
3. \( (y, 1) \)
4. \( (y, 2) \)
5. \( (z, 1) \)
6. \( (z, 2) \)

So, \( |A \times B| = 6 \).

### Step 4: Calculate the number of relations from A to B
The number of relations from set A to set B is equal to the number of subsets of the Cartesian product \( A \times B \). The number of subsets of a set with \( n \) elements is given by \( 2^n \).

Since we have \( n = 6 \) (the number of elements in \( A \times B \)), the number of relations is:
\[ 2^6 = 64 \]

### Final Answer
Thus, the number of relations from set A to set B is **64**.

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