GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 39

Question

For sets $A$ and $B$, let $f: A \rightarrow B$ and $g: B \rightarrow A$ be functions such that $f(g(x))=x$ for each $x \in B$. Which among the following statements is/are correct?

• A. The function $f$ must be one-to-one.
• B. The function $f$ must be onto.
• C. The function g must be one-to-one.
• D. The function $g$ must be onto.

Answer 1

The function $f$ must be onto and need not be one-to-one.

The function $g$ must be one-to-one and need not be onto.

For Example: Let $A = \{ 1,2 \}$, $B = \{ a \}$ ; Take $f(1) = a. f(2) = a, g(a) = 1$ ; So, $f $ need not be one-one and $g$ need not be onto.

Comments

Can u please explain in detail?

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@venkatesh pagadala, See the example that is added in the explanation now. 

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Here is the counter example of a and d

you can cleary see f is not one-one and g is not onto but still it satisfies f(g(x)) = x

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Sir, how mapping from (f(2) = a) is done as there is no mapping defined under g() of any element of set B to element "2" of set A ?