GATE CS 2023 Set 1 — Question 49
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Engineering Mathematics → Sets & Combinatorics → Sets, Relations & Functions
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Question
Let be an onto (or surjective) function, where and are nonempty sets. Define an equivalence relation on the set aswhere . Let be the set of all the equivalence classes under . Define a new mapping asWhich of the following statements is/are TRUE?
Correct answer
(B) F is an onto (or surjective) function.; (C) F is a one-to-one (or injective) function.; (D) F is a bijective function.
Solution
To determine which statements are true, we analyze the properties of the mapping defined by .
1.Well-definedness: A mapping is well-defined if for any , implies .
- By the definition of the equivalence relation , .
- Since and , we have .
- Thus, is well-defined. Statement (A) is false.
- We are given that is onto. Therefore, for any , there exists such that .
- For this , the equivalence class is in , and .
- Thus, is onto. Statement (B) is true.
- .
- By the definition of , .
- .
- Thus, is one-to-one. Statement (C) is true.
- Since is both one-to-one and onto, it is a bijective function. Statement (D) is true.
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