Answer to Question #129757 in Discrete Mathematics for gazal

Question

If f: A →B & g: B→C are two onto functions, then the mapping gof: A→C is also an
onto function. Prove

Answer 1

Definition : A function can be defined as an ONTO function if and only if

 all the elements belonging to the Codomain has its preimage in the Domain.

It is given to us that f : A→B and g : B→C where f & g are both

ONTO functions which can be understood as follows -

For example let us consider the following Domains & Codomains with their respective elements,

Given,

a1 is the preimage of b1, means f(a1) = b1

a2 is the preimage of b2, means f(a2) = b2

a3 is the preimage of b3, means f(a3) = b3

a4 is the preimage of b4, means f(a4) = b4

Similarly,

b1 is the preimage of c1, means g(b1) = c1

b2 is the preimage of c2, means g(b2) = c2

b3 is the preimage of c3, means g(b3) = c3

b4 is the preimage of c4, means g(b4) = c4

But,

b1 = f(a1) hence, g(b1) = g(f(a1)) = c1 similarly,

b2 = f(a2) hence, g(b2) = g(f(a1)) = c2

b3 = f(a3) hence, g(b3) = g(f(a1)) = c3

b4 = f(a4) hence, g(b4) = g(f(a1)) = c4

This means that for all the elements 'c' belonging to the Codomain 'C' there exists a preimage 'a' in the Domain 'A' which can be explained as under-

So, gof : A→C is also an ONTO function.