If A and B are two sets with `n(A)=4` and `n(B)=2` then the number of onto functions from A to B is

Options

A. 14
B. 16
C. 12
D. 15

Explanation

To find the number of onto functions from set A to set B, where \( n(A) = 4 \) and \( n(B) = 2 \), we can follow these steps:

### Step 1: Understand the Definitions
- An **onto function** (or surjective function) from set A to set B is a function where every element in B has at least one pre-image in A.
- The number of elements in set A is \( n(A) = 4 \).
- The number of elements in set B is \( n(B) = 2 \).

### Step 2: Calculate the Total Number of Functions
The total number of functions from set A to set B can be calculated using the formula:
\[ \text{Total functions} = n(B)^{n(A)} \]
Substituting the values:
\[ \text{Total functions} = 2^4 = 16 \]

### Step 3: Calculate the Number of Non-Onto Functions
To find the number of onto functions, we first need to calculate the number of non-onto functions. A non-onto function occurs when at least one element of B is not mapped by any element of A.

Since \( n(B) = 2 \), we can have two cases for non-onto functions:
1. Functions that map all elements of A to only one element of B.

For each element in B, there is one way to map all elements of A to that single element. Since there are 2 elements in B, the number of non-onto functions is:
\[ \text{Non-onto functions} = 2 \]

### Step 4: Calculate the Number of Onto Functions
Now, we can find the number of onto functions by subtracting the number of non-onto functions from the total number of functions:
\[ \text{Onto functions} = \text{Total functions} - \text{Non-onto functions} \]
Substituting the values:
\[ \text{Onto functions} = 16 - 2 = 14 \]

### Conclusion
Thus, the number of onto functions from set A to set B is \( 14 \).