Problem Definition
We need to find the number of functions $f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}$ that are not onto (not surjective).
- Domain set $A = \{1, 2, 3, 4\}$, so the size of the domain is $|A| = 4$.
- Codomain set $B = \{a, b, c\}$, so the size of the codomain is $|B| = 3$.
Total Number of Functions
The total number of possible functions from a set of size $m$ to a set of size $n$ is $n^m$. In this case, $m = 4$ and $n = 3$. Total functions = $3^4$. $ N_{total} = 3^4 = 81 $ There are 81 possible functions from $\{1, 2, 3, 4\}$ to $\{a, b, c\}$.
Number of Onto Functions
A function is onto if every element in the codomain is mapped to by at least one element from the domain.
The number of onto functions from a set of size $m$ to a set of size $n$ can be calculated using the principle of inclusion-exclusion:
$ N_{onto} = \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m $Substituting $m=4$ and $n=3$:
$ N_{onto} = \sum_{k=0}^{3} (-1)^k \binom{3}{k} (3-k)^4 $Calculating the terms:
- For $k=0$: $(-1)^0 \binom{3}{0} (3-0)^4 = 1 \cdot 1 \cdot 3^4 = 1 \cdot 81 = 81$
- For $k=1$: $(-1)^1 \binom{3}{1} (3-1)^4 = -1 \cdot 3 \cdot 2^4 = -3 \cdot 16 = -48$
- For $k=2$: $(-1)^2 \binom{3}{2} (3-2)^4 = 1 \cdot 3 \cdot 1^4 = 3 \cdot 1 = 3$
- For $k=3$: $(-1)^3 \binom{3}{3} (3-3)^4 = -1 \cdot 1 \cdot 0^4 = -1 \cdot 0 = 0
Summing the terms:
$ N_{onto} = 81 - 48 + 3 - 0 = 36 $ There are 36 onto functions from $\{1, 2, 3, 4\}$ to $\{a, b, c\}$.Number of Not Onto Functions
The number of functions that are not onto is the total number of functions minus the number of onto functions.
$ N_{\text{not onto}} = N_{total} - N_{onto} $ $ N_{\text{not onto}} = 81 - 36 $ $ N_{\text{not onto}} = 45 $ Therefore, there are 45 functions that are not onto.Similar Questions
- The number of points of discontinuity of the function $f (x) = \left[\frac{x^2}{2}\right] - [\sqrt{x}], x \in [0,4]$, where $[.]$denotes the greatest integer function, is ________
Let the function $f (x) = \frac{x}{3} + \frac{3}{x} + 3$, $x\neq0$ be strictly increasing in $(-\infty, a_1)\cup(a_2,\infty)$ and strictly decreasing in $(a_3, \alpha_4)\cup(a_4,a_5)$. Then $\sum_{i=1}^5 a_i^2$ is equal to
- $\lim_{x \to 0} \frac{xtan 2x-2x tan x}{(1-cos2x)^2}$ equals :-
Let $f(x) = \begin{cases} (x-1)^\frac{1}{2-x}, & x > 1, x \ne 2 \\ k, & x = 2 \end{cases}$ The value of k for which f is continuous at x = 2 is :
- Let $f: \mathbf{R} \to \mathbf{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbf{R}$ and $f'(a-1) = 0$, where $a$ is a real number. Let $g(x) = f(\tan^2 x - 2\tan x + a)$, $0 < x < \frac{\pi}{2}$.
Consider the following two statements :
(I) $g$ is increasing in $\left(0, \frac{\pi}{4}\right)$
(II) $g$ is decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
Then, - If the domain of the function $f(x) = \sin^{-1}\left(\frac{5-x}{3+2x}\right) + \frac{1}{\log_e(10-x)}$ is $(-\infty, \alpha] \cup [\beta, \gamma) - \{\delta\}$, then $6(\alpha + \beta + \gamma + \delta)$ is equal to
- Let the domain of the function $f(x) = \log_3 \log_5 \left(7 - \log_2 \left(x^2 - 10x + 85\right)\right) + \sin^{-1} \left(\left|\frac{3x - 7}{17 - x}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha + \beta$ is equal to :
- Let $[\cdot]$ denote the greatest integer function, and let $f(x) = \min \{\sqrt{2}x, x^2\}$.
Let $S = \{x \in (-2, 2) : \text{the function } g(x) = |x|[x^2] \text{ is discontinuous at } x\}$.
Then $\sum_{x \in S} f(x)$ equals - Let $f: \mathbb{R} \rightarrow (0, \infty)$ be a twice differentiable function such that $f(3) = 18, f'(3) = 0$ and $f''(3) = 4$. Then $\lim_{x \to 1} \left( \log_e \left( \frac{f(2+x)}{f(3)} \right)^{\frac{18}{(x-1)^2}} \right)$ is equal to :
- If the domain of the function $f(x) = \cos^{-1}\left( \frac{2x - 5}{11 - 3x} \right) + \sin^{-1}(2x^2 - 3x + 1)$ is the interval $[\alpha, \beta]$, then $\alpha + 2\beta$ is equal to :
Important Questions from Differential Calculus
- The number of points of discontinuity of the function $f (x) = \left[\frac{x^2}{2}\right] - [\sqrt{x}], x \in [0,4]$, where $[.]$denotes the greatest integer function, is ________
Let the function $f (x) = \frac{x}{3} + \frac{3}{x} + 3$, $x\neq0$ be strictly increasing in $(-\infty, a_1)\cup(a_2,\infty)$ and strictly decreasing in $(a_3, \alpha_4)\cup(a_4,a_5)$. Then $\sum_{i=1}^5 a_i^2$ is equal to
- $\lim_{x \to 0} \frac{xtan 2x-2x tan x}{(1-cos2x)^2}$ equals :-
Let $f(x) = \begin{cases} (x-1)^\frac{1}{2-x}, & x > 1, x \ne 2 \\ k, & x = 2 \end{cases}$ The value of k for which f is continuous at x = 2 is :
- Let $f: \mathbf{R} \to \mathbf{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbf{R}$ and $f'(a-1) = 0$, where $a$ is a real number. Let $g(x) = f(\tan^2 x - 2\tan x + a)$, $0 < x < \frac{\pi}{2}$.
Consider the following two statements :
(I) $g$ is increasing in $\left(0, \frac{\pi}{4}\right)$
(II) $g$ is decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
Then,
