The function f : Z (INTEGERS) -> N(NATURAL NUMBER) U {0} is defined by
f(n) =
(2n) if n >= 0,
−(2n + 1) if n < 0.
Prove f is a bijection.
I know its simple, but i need to show it by using lots of math and little english
Hello neelpatel89Let me set out for you what you need to prove.
You need to prove that is (a) an injection (one-to-one); and (b) a surjection (onto function).
For (a), you need to show that if and , then . You'll need to consider the cases where and take various combinations of positive and negative signs.
To start you off, suppose , and . Then ...?
Then look at what happens if and are both negative. Finally show that if and have different signs, then .
For (b) you need to show that for any , we can find an such that . Again you'll need to consider two separate cases: this time it will be whether is even or odd.
Let us know if you can't see how to complete it from here.
Grandad
Hello neelpatel89
Here's the proof, then, using the structure in my earlier post.
(a) Consider , where
First: and
Then and
Finally, we note that if is even and if , is odd. So if and have different signs.
So for all , and therefore is injective.
(b) Consider .
Then if is even, for some . Also . So . Therefore for some .
If is odd, and therefore for some
, where
Therefore for all , there is an such that . Therefore is surjective.
Grandad