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If a relation f:X→Y is a function , then for g:Y→X to be a function ,function f need to be
- a)one-one and onto
- b)one-one
- c)onto but not one-one
- d)many one
Correct answer is option 'A'. Can you explain this answer?
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If a relation f:X→Y is a function , then for g:Y→X to be a...
Understanding the Relation Between Functions
When dealing with two functions, f: X → Y and g: Y → X, the properties of f directly impact whether g can be defined as a function.
Function Definition
- A function f maps each element in the set X to exactly one element in the set Y.
- For g to qualify as a function, it also needs to assign each element of Y to exactly one element of X.
Requirements for g to be a Function
To ensure that g is a well-defined function, f must be both one-one and onto. Here's why:
1. One-One (Injective)
- If f is one-one, each element of Y corresponds to a unique element in X.
- This uniqueness ensures that g can map each element of Y back to its original element in X without any ambiguity.
2. Onto (Surjective)
- If f is onto, every element in Y is covered by at least one element in X.
- This means that for every y in Y, there exists an x in X such that f(x) = y, allowing g to map y back to its corresponding x.
Conclusion
- If f is both one-one and onto, g can successfully reverse the mapping of f.
- If f is only one-one, some elements in Y may not have a preimage in X, making g undefined for those elements.
- If f is only onto, multiple elements in X may map to the same element in Y, leading to ambiguity in defining g.
Thus, for g to be a valid function, f must be both one-one and onto, making option 'A' the correct answer.
When dealing with two functions, f: X → Y and g: Y → X, the properties of f directly impact whether g can be defined as a function.
Function Definition
- A function f maps each element in the set X to exactly one element in the set Y.
- For g to qualify as a function, it also needs to assign each element of Y to exactly one element of X.
Requirements for g to be a Function
To ensure that g is a well-defined function, f must be both one-one and onto. Here's why:
1. One-One (Injective)
- If f is one-one, each element of Y corresponds to a unique element in X.
- This uniqueness ensures that g can map each element of Y back to its original element in X without any ambiguity.
2. Onto (Surjective)
- If f is onto, every element in Y is covered by at least one element in X.
- This means that for every y in Y, there exists an x in X such that f(x) = y, allowing g to map y back to its corresponding x.
Conclusion
- If f is both one-one and onto, g can successfully reverse the mapping of f.
- If f is only one-one, some elements in Y may not have a preimage in X, making g undefined for those elements.
- If f is only onto, multiple elements in X may map to the same element in Y, leading to ambiguity in defining g.
Thus, for g to be a valid function, f must be both one-one and onto, making option 'A' the correct answer.
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If a relation f:X→Y is a function , then for g:Y→X to be a...
A) one-one and onto, right answer...
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If a relation f:X→Y is a function , then for g:Y→X to be a function ,function f need to bea)one-one and ontob)one-onec)onto but not one-oned)many oneCorrect answer is option 'A'. Can you explain this answer?
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