Question

Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{a, b, c\}$ then total number of functions from $A$ to $B$ which are not onto are

• A. 84
• B. 93
• C. 100
• D. 54

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A function is "onto" (surjective) if every element in the codomain has at least one pre-image in the domain.
To find "not onto" functions, we subtract onto functions from total possible functions.
Step 2: Key Formula or Approach:
1. Total functions $n^m$, where $n = |B|$ and $m = |A|$.
2. Number of onto functions from set of size $m$ to set of size $n$ is given by $n! \cdot S(m, n)$, where $S$ is Stirling number of the second kind, or by Inclusion-Exclusion.
Step 3: Detailed Explanation:
1. Total functions from $A$ to $B$:
$|B|^{|A|} = 3^5 = 243$.
2. Calculate onto functions using the formula: $n^m - \binom{n}{1}(n-1)^m + \binom{n}{2}(n-2)^m$:
For $n=3, m=5$:
$3^5 - \binom{3}{1}2^5 + \binom{3}{2}1^5 = 243 - 3(32) + 3(1) = 243 - 96 + 3 = 150$.
3. Calculate functions which are not onto:
Not onto $= \text{Total} - \text{Onto} = 243 - 150 = 93$.
Step 4: Final Answer:
The total number of functions that are not onto is 93.