GATE CS 2025 Set 1 — Question 17
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Engineering Mathematics → Sets & Combinatorics → Sets, Relations & Functions
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Question
is a function from to , is a function from to , and their composition defined as is a mapping from to .If and are onto (surjective) functions, which ONE of the following is TRUE about the function ?
Correct answer
(D) g(·) is not required to be a one-to-one or onto function.
Solution
Let and . The composition is .
- Given is onto, for every , there exists such that .
- This implies that for every , there exists such that . Thus must be onto (which is already given).
- However, does not need to be onto. For example, if , , , , and . Here and are onto, but is not onto because is not in its range.
- Similarly, does not need to be one-to-one. Multiple elements in could map to the same element in without affecting the surjectivity of the composition.
- Therefore, is not required to be one-to-one or onto. Option (D) is correct.
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