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 sizes
Let \( A = \{1, 2, 3\} \) and \( B = \{a, b\} \).
- The number of elements in set \( A \) (denoted as \( M \)) is 3.
- The number of elements in set \( B \) (denoted as \( N \)) is 2.

### 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:
\[ \text{Total relations} = 2^{M \times N} \]
Here, \( M \times N \) is the product of the number of elements in both sets.

Substituting the values:
\[ M \times N = 3 \times 2 = 6 \]
Thus, the total number of relations is:
\[ 2^{6} = 64 \]

### Step 3: Calculate the number of non-empty relations
To find the number of non-empty relations, we subtract the empty relation from the total number of relations:
\[ \text{Non-empty relations} = \text{Total relations} - 1 \]
So, we have:
\[ \text{Non-empty relations} = 64 - 1 = 63 \]

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

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