If A={a,y} and B={2,5} , then find the number of relations from set A to B.

Options

A. 4
B. 8
C. 16
D. 32

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:
- Set A = {a, y}
- Set B = {2, 5}

### Step 2: Determine the number of elements in each set
- The number of elements in set A (|A|) = 2 (elements: a, y)
- The number of elements in set B (|B|) = 2 (elements: 2, 5)

### Step 3: Calculate the Cartesian product A × B
The Cartesian product A × B consists of all possible ordered pairs (a, b) where a ∈ A and b ∈ B. Thus, we can list the pairs:
- (a, 2)
- (a, 5)
- (y, 2)
- (y, 5)

So, A × B = {(a, 2), (a, 5), (y, 2), (y, 5)}.

### Step 4: Count the number of elements in A × B
The number of elements in A × B is 4.

### Step 5: Determine the number of subsets of A × B
The number of relations from set A to set B is equal to the number of subsets of A × B. The number of subsets of a set with n elements is given by 2^n.

Since |A × B| = 4, the number of subsets is:
Number of subsets = 2^4 = 16.

### Conclusion
The total number of relations from set A to set B is 16.

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