If f : A → B and g : B → C are onto functions, show that gof is a onto function.


If f : A → B and g : B → C are onto functions, show that gof is a onto function.


Given,  f : A → B and g : B → C are onto.

Then, gof : A → C

Let us take an element z in the co-domain (C).

Now, $z$ is in $C$ and $g: B \rightarrow C$ is onto.

So, there exists some element $y$ in $B$, such that $g(y)=z$

Now, $y$ is in $B$ and $f: A \rightarrow B$ is onto.

So, there exists some $x$ in $A$, such that $f(x)=y \ldots$ (2)

From (1) and (2),

$z=g(y)=g(f(x))=(g \circ f)(x)$

So, $z=(g \circ f)(x)$, where $x$ is in $A$.

Hence, gof is onto.




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