Question

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

Answer 1

Answer:

Given sets E={1,2,3,4} and F={1,2}, how many functions E->F are possible? Well, each element of E could be mapped to 1 of 2 elements of F, therefore the total number of possible functions E->F is 2*2*2*2 = 16.

How many are “onto”? All but 2. The only two functions which are not “onto” would be these two:

y=f(x)=1 (for any x)

y=g(x)=2 (for any x)

(In each of those two cases, “x” means “any domain element” and “y” means “that codomain element to which the function maps x”. For example, if x is 3, then f(x)=1 and g(x)=2.)

The reason those are the only possible non-onto functions is that the other 14 possible functions would all map elements of E to both elements of the F, so that range=codomain=F, so they’re “onto” (surjective). (To be non-onto, a function would have to leave one or more codomain elements not-mapped-to. A classic example of a non-onto function is sin:R->R, which leaves all elements of the codomain outside the closed interval [-1,1] not-mapped-to.)

Therefore, the answer is: 14 surjective (“onto”) and 2 non-surjective (non-“onto”) functions possible; however, no injective (“one-to-one”) or bijective (“one-to-one-and-onto”) functions are possible, because the domain has greater cardinality than the codomain.

Answer 2

Given,

A= {a,b,c} and B = {1,2,3,4}

To find,

Number of functions from A to B which are not onto.

Solution,

The number of onto function from A to B is the coefficient of

$x^{4}$ in  $4![x + \frac{x^{2} }{2!} + \frac{x^{3} }{3!} + ...]^3$ , that is

coefficient of $x^{4}$ in 4![$e^{3x}$ - 3$e^{2x}$ + 3$e^{x}$ - 1]

                           =  $4![ \frac{3^{4} }{4!} + \frac{3.2^{4} }{4!} + \frac{3}{4!} ]$

                           = $3^{4}$ - 3.$2^{4}$ + 3

                           = 81 - 48 + 3

                           = 36

Total number of functions = $A^{B}$ = $3^{4}$ = 81

Number of functions not onto = 81 - 36 = 45

Hence, the number of functions from A to B which are not onto are 45.