Explicit formula

  • Thread starter persian52
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  • #1
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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[itex]_{2}[/itex](n)= ⌊[itex]\frac{n}{3}[/itex]⌋. it's onto cause f[itex]_{2}[/itex](3n)= n for every n. but, it's not one-to-one, cause f[itex]_{2}[/itex](1)= ⌊[itex]\frac{1}{3}[/itex]⌋ = 0 = f[itex]_{2}[/itex](0)

b) f[itex]_{4}[/itex](n)=4 ⌊[itex]\frac{n}{4}[/itex]⌋. This is not one-to-one, cause it's counterexample f[itex]_{4}[/itex](0) = f(1) = 0. Nor is it onto cause there is no odd number in the range of f[itex]_{4}[/itex]
 
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Answers and Replies

  • #2
Dick
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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.
 
  • #3
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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.

Sorry for that, it wasn't completed yet, now it's.
 
  • #4
Dick
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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.
 
  • #5
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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.
 
  • #6
eumyang
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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.
 
  • #7
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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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