Maths · One-one, into and onto functions
The function defined by =\boldsymbol{x}-\mathbf{5}\left[\frac{\boldsymbol{x}}{\m
The function \( \boldsymbol{f}: \boldsymbol{N} \rightarrow \boldsymbol{N} \) defined by \( \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}-\mathbf{5}\left[\frac{\boldsymbol{x}}{\mathbf{5}}\right], \) where \( \mathrm{N} \) is the set of natural numbers and \( [x] \) denotes the greatest integer less than or equal to \( x \) is
- A. one-one but not onto
- B. neither one-one nor onto
- C. onto but not one-one
- D. one-one and onto
Step-by-step solution
The function f(x)=x-5⌊x/5⌋ gives the remainder when x is divided by 5, so its range is {0,1,2,3,4} (or without 0 if N starts at 1). Since many natural numbers map to the same remainder (e.g., f(1)=f(6)=1), it is not one-one. Also, natural numbers greater than 4 are not in the range, so it is not onto.
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