document.write( "Question 548064: Prove that the function is bijective. The function is f: N--->Z f(x)=(1+(−1)^x * (2x − 1)) /4\r
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Algebra.Com's Answer #356755 by richard1234(7193) $\"\"$ $\"About$

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Presuming\r
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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.\r
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\n" ); document.write( "\n" ); document.write( "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.\r
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