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(B) Given that $g \circ f: A \rightarrow C$ is a bijection,it is both injective (one-one) and surjective (onto). $1$. For injectivity: Let $x_1, x_2 \

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$f$ is an injection and $g$ is a surjection

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(B) Given that $g \circ f: A \rightarrow C$ is a bijection,it is both injective (one-one) and surjective (onto). $1$. For injectivity: Let $x_1, x_2 \in A$ such that $f(x_1) = f(x_2)$. Then $g(f(x_1)) = g(f(x_2))$,which implies $(g \circ f)(x_1) = (g \circ f)(x_2)$. Since $g \circ f$ is injective,$x_1 = x_2$. Thus,$f$ must be an injection. $2$. For surjectivity: Since $g \circ f$ is surjective,for every $z \in C$,there exists an $x \in A$ such that $(g \circ f)(x) = z$. This can be written as $g(f(x)) = z$. Since $f(x) \in B$,there exists an element $b = f(x) \in B$ such that $g(b) = z$. Thus,for every $z \in C$,there exists $b \in B$ such that $g(b) = z$. Therefore,$g$ is a surjection. Hence,$f$ is an injection and $g$ is a surjection.

If $f: A \rightarrow B$ and $g: B \rightarrow C$ are two functions such that $g \circ f: A \rightarrow C$ is a bijection,then which one of the following is always true?

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