Let `A={1,2,3} and B ={a,b}`
what is the number of non empty relations from A to B

Answer

To find the number of non-empty relations from set \( A \) to set \( B \), we can follow these steps:

### Step 1: Identify the sets and their elements
We have:
- Set \( A = \{1, 2, 3\} \) which has 3 elements.
- Set \( B = \{a, b\} \) which has 2 elements.

### Step 2: Calculate the total number of relations
The total number of relations from set \( A \) to set \( B \) can be calculated using the formula:
\[ \text{Total relations} = 2^{m \times n} \]
where \( m \) is the number of elements in set \( A \) and \( n \) is the number of elements in set \( B \).

Here, \( m = 3 \) and \( n = 2 \).

So, we calculate:
\[ \text{Total relations} = 2^{3 \times 2} = 2^6 = 64 \]

### Step 3: Calculate the number of non-empty relations
Since we want the number of non-empty relations, we need to subtract the empty relation from the total relations calculated in Step 2:
\[ \text{Non-empty relations} = \text{Total relations} - 1 \]
Thus,
\[ \text{Non-empty relations} = 64 - 1 = 63 \]

### Final Answer
The number of non-empty relations from set \( A \) to set \( B \) is \( 63 \).

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