Explicit Formula for Onto/Non-One-to-One Function

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An explicit formula for an onto but not one-to-one function is f_2(n) = ⌊n/3⌋, as it maps every n to a unique output while allowing multiple inputs to yield the same result. For a function that is neither one-to-one nor onto, f_4(n) = 4⌊n/4⌋ was proposed, which fails to cover all natural numbers and has overlapping outputs. There was confusion regarding the definition of natural numbers, with some participants arguing that zero should not be included. This led to a discussion about the varying definitions of natural numbers versus whole numbers. The conversation highlighted the need for clarity in mathematical definitions to avoid misunderstandings.
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Homework Statement


Give an explicit formula for a function f : ℕ ⇒ ℕ that is

a) onto, but not one-to-one.
b) neither one-to-one nor onto.


1. The attempt at a solution
a) The formula f_{2}(n)= ⌊\frac{n}{3}⌋. it's onto cause f_{2}(3n)= n for every n. but, it's not one-to-one, cause f_{2}(1)= ⌊\frac{1}{3}⌋ = 0 = f_{2}(0)

b) f_{4}(n)=4 ⌊\frac{n}{4}⌋. This is not one-to-one, cause it's counterexample f_{4}(0) = f(1) = 0. Nor is it onto cause there is no odd number in the range of f_{4}
 
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1/3 isn't equal to 0. You have to explain what f_2 means a lot better than that. Do you mean something like the floor function? Oh, I see you do. Guess I should wait till you finish posting.
 
Dick said:
1/3 isn't equal to 0. You have to explain what f_2 means a lot better than that. Do you mean something like the floor function? Oh, I see you do. Guess I should wait till you finish posting.

Sorry for that, it wasn't completed yet, now it's.
 
persian52 said:
Sorry for that, it wasn't completed yet, now it's.

Mmm. ##f_2(1)=0##. 0 isn't in N is it? You might have to modify it a bit.
 
Dick said:
Mmm. ##f_2(1)=0##. 0 isn't in N is it? You might have to modify it a bit.

natural numbers { 0, 1, 2, 3, ...}

Yes it's.
 
persian52 said:
natural numbers { 0, 1, 2, 3, ...}

Yes it's.
Not to me it isn't. In some places natural numbers do not include zero {1, 2, 3, ...}. We use another term, whole numbers, to indicate {0, 1, 2, 3, ...}. Why is there no agreement on the definition of natural numbers I have no idea.
 
eumyang said:
Not to me it isn't. In some places natural numbers do not include zero {1, 2, 3, ...}. We use another term, whole numbers, to indicate {0, 1, 2, 3, ...}. Why is there no agreement on the definition of natural numbers I have no idea.

i agree.
 
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