The number of functions f:1,2,3,4 a,b,c, which are not onto, is :

Question

The number of functions $f:{1,2,3,4}	o {a,b,c}$, which are not onto, is :

Accepted Answer

Total number of functions from ${1,2,3,4}$ to ${a,b,c}$ is

${3}^{4}=81$

Now we find the number of onto functions.

For a function to be onto, all the three elements $a,b,c$ must appear as images of elements of ${1,2,3,4}$.

Using inclusion-exclusion principle:

Total functions: ${3}^{4}=81$

Let

$A$ = functions missing $a$

$B$ = functions missing $b$

$C$ = functions missing $c$

If one element is missing, then the function goes into only $2$ elements.

So,

$|A|=|B|=|C|={2}^{4}=16$

Hence,

$|A|+|B|+|C|=3	imes 16=48$

If two elements are missing, then all $4$ elements of domain must go to only one element of codomain.

So,

$|A\cap B|=|B\cap C|=|C\cap A|={1}^{4}=1$

Thus,

$|A\cap B|+|B\cap C|+|C\cap A|=3$

If all three are missing, that is impossible, so

$|A\cap B\cap C|=0$

Therefore, number of functions which are not onto is

$|A\cup B\cup C|=48-3=45$

So the correct answer is

$45$

Hence, Option B is correct.

Solutions · 1