Question 548064: Prove that the function is bijective. The function is f: N--->Z f(x)=(1+(−1)^x * (2x − 1)) /4
I tried to attempt it using induction. I did a base case for subjectivity of f(1) and f(2) and it worked. Then I tried to substitute k+1 for x but I have gotten stuck.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Presuming
 = \frac{1 + (-1)^x*(2x - 1)}{4}) where x is a positive integer. We know that the domain is {1,2,3,4,...} and the range is {...-2,-1,0,1,2,...}. Hence we can say that if x is even, the domain is {2,4,...} and the range is {1,2,3,...} (since the function is equivalent to f(x) = x/2). If x is odd, the domain is {1,3,...} and the range is {0,-1,-2,...}. There exists a one-to-one correlation between both disjoint domains and ranges, so the entire function is bijective.
However, we must be careful with one-to-one correlations with infinite sets such as {2,4,6,8,...} and {1,2,3,...} if x is even (e.g. look up "Hilbert's paradox of the Grand Hotel"). We can still define a function (e.g. f(x) = x/2) so we don't have to worry about this.
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