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The function f : N → N defined by f(x) = x-5 [x/5], where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :
- a)one-one and onto
- b)one-one but not onto
- c)onto but not one-one
- d)neither one-one nor onto
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The functionf : N → N defined by f(x) = x-5 [x/5],where N is the ...
Explanation:
One-one:
- To check if the function is one-one, we need to see if each element in the domain maps to a unique element in the codomain.
- Let's consider two natural numbers x and y such that f(x) = f(y).
- This implies x-5 [x/5] = y-5 [y/5].
- For the function to be one-one, x must be equal to y. However, this is not always the case as different natural numbers can map to the same value.
Onto:
- To check if the function is onto, we need to see if every element in the codomain has a pre-image in the domain.
- If we consider the natural number k in the codomain, we need to find a natural number in the domain such that f(x) = k.
- However, due to the floor function [x/5], not every natural number has a pre-image in the domain, making the function not onto.
Therefore, the function f : N → N defined by f(x) = x-5 [x/5] is neither one-one nor onto, and the correct answer is option 'D'.
One-one:
- To check if the function is one-one, we need to see if each element in the domain maps to a unique element in the codomain.
- Let's consider two natural numbers x and y such that f(x) = f(y).
- This implies x-5 [x/5] = y-5 [y/5].
- For the function to be one-one, x must be equal to y. However, this is not always the case as different natural numbers can map to the same value.
Onto:
- To check if the function is onto, we need to see if every element in the codomain has a pre-image in the domain.
- If we consider the natural number k in the codomain, we need to find a natural number in the domain such that f(x) = k.
- However, due to the floor function [x/5], not every natural number has a pre-image in the domain, making the function not onto.
Therefore, the function f : N → N defined by f(x) = x-5 [x/5] is neither one-one nor onto, and the correct answer is option 'D'.
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The functionf : N → N defined by f(x) = x-5 [x/5],where N is the set of natural numbers and[x] denotes the greatest integer less than or equal to x, is :a)one-one and ontob)one-one but not ontoc)onto but not one-oned)neither one-one nor ontoCorrect answer is option 'D'. Can you explain this answer?
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