Are f and g both necessarily onto,if g circ f | vedclass

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Question

(C) If $g \circ f$ is onto,then $g$ must be onto,but $f$ does not necessarily have to be onto.Consider $f: \{1, 2, 3, 4\} \rightarrow \{1, 2, 3, 4\}$

Vedclass · Accepted Answer

No,only $g$ must be onto.

Vedclass · Answer

(C) If $g \circ f$ is onto,then $g$ must be onto,but $f$ does not necessarily have to be onto.Consider $f: \{1, 2, 3, 4\} \rightarrow \{1, 2, 3, 4\}$ and $g: \{1, 2, 3, 4\} \rightarrow \{1, 2, 3\}$ defined as:$f(1) = 1, f(2) = 2, f(3) = 3, f(4) = 3$$g(1) = 1, g(2) = 2, g(3) = 3, g(4) = 3$Here,the range of $g \circ f$ is $\{1, 2, 3\}$,which is equal to the codomain of $g$,so $g \circ f$ is onto.However,the range of $f$ is $\{1, 2, 3\}$,which is not equal to its codomain $\{1, 2, 3, 4\}$.Thus,$f$ is not onto.

Are $f$ and $g$ both necessarily onto,if $g \circ f$ is onto?

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