If A = {a,b) and B = {x, y, z), then the number of relations from B to A is

Options

A. 8
B. 16
C. 32
D. 64

Answer

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

### Step 1: Identify the sets
Given:
- Set A = {a, b}
- Set B = {x, y, z}

### Step 2: Find the Cartesian Product B × A
The Cartesian product B × A consists of all possible ordered pairs (b, a) where b is an element from set B and a is an element from set A.

Calculating B × A:
- From element x in B: (x, a), (x, b)
- From element y in B: (y, a), (y, b)
- From element z in B: (z, a), (z, b)

Thus, B × A = {(x, a), (x, b), (y, a), (y, b), (z, a), (z, b)}

### Step 3: Count the number of elements in B × A
Now, we count the number of ordered pairs in B × A:
- There are 6 elements in B × A:
1. (x, a)
2. (x, b)
3. (y, a)
4. (y, b)
5. (z, a)
6. (z, b)

So, the number of elements in B × A is 6.

### Step 4: Calculate the number of relations from B to A
The number of relations from set B to set A is equal to the number of subsets of B × A. The number of subsets of a set with n elements is given by 2^n.

Since we have 6 elements in B × A, the number of relations from B to A is:
\[ 2^{|B \times A|} = 2^6 \]

### Step 5: Calculate 2^6
Calculating \(2^6\):
\[ 2^6 = 64 \]

### Conclusion
Therefore, the number of relations from set B to set A is 64.

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