Let A = {1, 2, 3, 4} and B = {a, b}. A function f : A `to` B is selected randomly. Probability that function is an onto function is

Options

A. `1/8`
B. `5/8`
C. `7/8`
D. `3/8`

Answer

To solve the problem, we need to find the probability that a randomly selected function \( f: A \to B \) is an onto function, where \( A = \{1, 2, 3, 4\} \) and \( B = \{a, b\} \).

### Step 1: Calculate the total number of functions from \( A \) to \( B \)

Each element in set \( A \) can be mapped to either of the two elements in set \( B \). Since there are 4 elements in \( A \) and each can be mapped independently to 2 elements in \( B \), the total number of functions is calculated as follows:

\[ \text{Total functions} = 2^4 = 16 \]

**Hint:** Remember that for each element in set \( A \), you have multiple choices in set \( B \).

### Step 2: Determine the conditions for a function to be onto

A function is onto (or surjective) if every element in the codomain \( B \) has at least one pre-image in the domain \( A \). For our function \( f: A \to B \) to be onto, both \( a \) and \( b \) must be used at least once in the mapping.

### Step 3: Calculate the number of onto functions

To find the number of onto functions, we can use the principle of complementary counting. We will first count the functions that are not onto.

1. **Case 1:** All elements in \( A \) map to \( a \). This gives us 1 function: \( f(1) = a, f(2) = a, f(3) = a, f(4) = a \).
2. **Case 2:** All elements in \( A \) map to \( b \). This also gives us 1 function: \( f(1) = b, f(2) = b, f(3) = b, f(4) = b \).

Thus, the total number of functions that are not onto is:

\[ \text{Not onto functions} = 1 + 1 = 2 \]

### Step 4: Calculate the number of onto functions

Now, we can find the number of onto functions by subtracting the number of not onto functions from the total number of functions:

\[ \text{Onto functions} = \text{Total functions} - \text{Not onto functions} = 16 - 2 = 14 \]

### Step 5: Calculate the probability of selecting an onto function

The probability \( P \) that a randomly selected function is onto is given by the ratio of the number of onto functions to the total number of functions:

\[ P(\text{onto}) = \frac{\text{Onto functions}}{\text{Total functions}} = \frac{14}{16} = \frac{7}{8} \]

### Final Answer

Thus, the probability that the function is an onto function is:

\[ \boxed{\frac{7}{8}} \]