If `P=(a,b,c) and Q=(1,2)`, then the total number of relations P to Q are not functions is

Options

A. 56
B. 8
C. 9
D. 55

Explanation

To solve the problem, we need to determine the total number of relations from set P to set Q that are not functions. Here are the steps to arrive at the solution:

### Step 1: Identify the sets
We have:
- Set P = {a, b, c} which has 3 elements.
- Set Q = {1, 2} which has 2 elements.

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

Here, \( m = 3 \) and \( n = 2 \):
\[ \text{Total Relations} = 2^{(3 \times 2)} = 2^6 = 64 \]

### Step 3: Calculate the total number of functions from P to Q
A function from set P to set Q must assign each element in P to exactly one element in Q. Each element in P has 2 choices (either 1 or 2 from Q).

Thus, the total number of functions is:
\[ \text{Total Functions} = n^m = 2^3 = 8 \]

### Step 4: Calculate the number of relations that are not functions
To find the number of relations that are not functions, we subtract the number of functions from the total number of relations:
\[ \text{Relations that are not functions} = \text{Total Relations} - \text{Total Functions} \]
\[ = 64 - 8 = 56 \]

### Conclusion
The total number of relations from P to Q that are not functions is **56**.

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