Is There a Function f: Z -> Z Such That f(f(n))=-n for Every Integer?

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In summary, the conversation discusses the existence of a function f: Z -> Z such that f(f(n)) = -n for all integers n. The participants suggest various ideas, but no one can prove or disprove the possibility of such a function. One participant suggests a piecewise permutation function that satisfies the given condition, but it is not integer-valued. Overall, it is believed that such a function does not exist.
  • #1
ypatia
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Is there any function (if any) f: Z -> Z such that
f(f(n))=-n , for every n belongs to Z(integers) ??


I think that there is not any function like the one described above but how can we prove it. Any ideas??
Thanks in Advance
 
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  • #2
How about f(n)=in?
 
  • #3
Not integer-valued. (I assume if the OP meant Gaussian integers that would have been mentioned, since that's the obvious solution.)

I've been thinking about this for a few hours now and I can't see any way to do it, but I can't prove that it's impossible.
 
  • #4
How about

for n>0
f(2n-1)=2n
f(2n)=-2n+1
f(-2n+1)=-2n
f(-2n)=2n-1

f(0)=0
 
  • #5
Nice, chronon. Nice.
 
  • #6
Indeed - it's nice to visualize f as a piecewise permutation

(0)(-2,-1,2,1)(-4,-3,4,3)...(-2n,-2n+1,2n,2n-1)...

and recall that (abcd)^2=(ac)(bd)
 

FAQ: Is There a Function f: Z -> Z Such That f(f(n))=-n for Every Integer?

What is an integer-valued function?

An integer-valued function is a mathematical function that takes in one or more input values and produces an integer as its output. This means that the values of the function are restricted to whole numbers.

How is an integer-valued function different from a real-valued function?

An integer-valued function only produces integer values as its output, while a real-valued function can produce any real number as its output. Additionally, the domain of an integer-valued function is typically restricted to whole numbers, while the domain of a real-valued function can be any real number.

What are some common examples of integer-valued functions?

Some common examples of integer-valued functions include counting functions, such as the number of students in a classroom or the number of items in a set. Other examples include rounding functions, such as rounding to the nearest whole number or rounding down to the nearest integer.

Can an integer-valued function have more than one input variable?

Yes, an integer-valued function can have more than one input variable. For example, a function that calculates the number of ways to choose a certain number of objects from a set could have two input variables: the number of objects in the set and the number of objects chosen.

How are integer-valued functions used in real life?

Integer-valued functions are used in various fields such as mathematics, computer science, and economics. In mathematics, they are used to model discrete phenomena such as counting and combinatorics. In computer science, they are used to optimize algorithms and data structures by analyzing the integer values they produce. In economics, they are used to model discrete quantities such as the number of units produced or the number of customers served.

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