example of a function (one to one & onto)

TheRekz · #1 $Old$ July 11th, 2007, 09:20 AM

Give an example of a function from N to N that is one to one but not onto

My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 3, f(b) = 4, f(c) = 1. Is this the correct example to this question? What does it mean from N to N?

ThePerfectHacker · #2 $Old$ July 11th, 2007, 11:08 AM

Quote:

Originally Posted by TheRekz $View Post$

Give an example of a function from N to N that is one to one but not onto

My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 3, f(b) = 4, f(c) = 1. Is this the correct example to this question? What does it mean from N to N?

f(1)=2, f(2)=3, f(3)=4, ... f(n) = n+1, ...

Plato · #3 $Old$ July 11th, 2007, 11:27 AM

Quote:

Originally Posted by TheRekz $View Post$

Give an example of a function from N to N that is one to one but not onto
My answer is the function from {a,b,c} to {1,2,3,4} with f(a) = 3, f(b) = 4, f(c) = 1. Is this the correct example to this question? What does it mean from N to N?

No it is not correct.
First N is the set of counting numbers either {0,1,2,3,...} or {1,2,3,4,...} (i.e. it contains 0 or not depending on your textbook). So N is infinite; so your example must be infinite also. Here is another example in addition to the one given above.
$f:N \mapsto N,\quad \left( {n \in N} \right)\left[ {f(n) = 2^n } \right]\quad$

ThePerfectHacker · #4 $Old$ July 11th, 2007, 11:43 AM

Note if $S$ is a finite set it is NOT POSSIBLE to find:

$f:S\to S$ that is one-to-one but not onto. This is a consequence of the Pigeonhole Principle. This is a distinction between finite and infinite sets.

TheRekz · #5 $Old$ July 11th, 2007, 12:34 PM

Quote:

Originally Posted by Plato $View Post$

No it is not correct.
First N is the set of counting numbers either {0,1,2,3,...} or {1,2,3,4,...} (i.e. it contains 0 or not depending on your textbook). So N is infinite; so your example must be infinite also. Here is another example in addition to the one given above.
$f:N \mapsto N,\quad \left( {n \in N} \right)\left[ {f(n) = 2^n } \right]\quad$

so what you're saying is that the function f(x) = 2^x will work as an example of this question?

ThePerfectHacker · #6 $Old$ July 11th, 2007, 12:36 PM

Quote:

Originally Posted by TheRekz $View Post$

so what you're saying is that the function f(x) = 2^x will work as an example of this question?

Yes. The function I gave it even simpler.

TheRekz · #7 $Old$ July 11th, 2007, 06:29 PM

What if the question is neither one to one nor onto? What would be a perfect example? Would x^2 + 1 work?

ThePerfectHacker · #8 $Old$ July 11th, 2007, 07:10 PM

Quote:

Originally Posted by TheRekz $View Post$

What if the question is neither one to one nor onto? What would be a perfect example? Would x^2 + 1 work?

x^2+1 is one-to-one in this case.

How about,
f(1)=1
f(2)=1
f(3)=1
f(4)=1
...
f(n)=1

Plato · #9 $Old$ July 11th, 2007, 07:19 PM

Quote:

Originally Posted by TheRekz $View Post$

What if the question is neither one to one nor onto? What would be a perfect example? Would x^2 + 1 work?

No! Because the elements of N are non-negative that function is one-to-one.
Look at the function $f:N \mapsto N,\quad f(n) = \text{floor}\left( {\frac{{n + 5}}{2}} \right)$

TheRekz · #10 $Old$ July 11th, 2007, 07:40 PM

can you give me other examples that does not use the floor operator?

TheRekz · #11 $Old$ July 12th, 2007, 09:58 AM

N is a set of natural numbers right? And a natural number can't be negative?

Jhevon · #12 $Old$ July 12th, 2007, 10:14 AM

Quote:

Originally Posted by TheRekz $View Post$

N is a set of natural numbers right? And a natural number can't be negative?

correct. the natural numbers is the set of positive integers: 1,2,3,4,5,6....

wow, i could actually answer a question in this thread

Plato · #13 $Old$ July 12th, 2007, 10:29 AM

Quote:

Originally Posted by TheRekz $View Post$

N is a set of natural numbers right? And a natural number can't be negative?

Look! We do not know what textbook or set of notes you are following.
Sorry to say, but it is true nonetheless, there are no hard and fast definition in mathematics. So you read the definition of $N$ in your text material. Most texts that I have seen require $N$ to be either {0,1,2,3,…} or {1,2,3,…}*. So yes the customary and usual definitions of $N$ mean that the set is made of non-negative integers.

* See this site: Counting Number -- from Wolfram MathWorld