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

[persian52]

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}

 

[Dick]

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.

 

[persian52]

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.

 

[Dick]

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.

 

[persian52]

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.

 

[eumyang]

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.

 

[persian52]

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.