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Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i))=I, for all 0 ≤ i ≤ 2014. Consider the following statements: P. For each such function it must be the case that for every i, f(i) = i Q. For each such function it must be the case that for some i, f(i) = i R. Each such function must be onto. Which one of the following is CORRECT?

Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i))=I, for all 0 ≤ i ≤ 2014. Consider the following statements: P. For each such function it must be the case that for every i, f(i) = i Q. For each such function it must be the case that for some i, f(i) = i R. Each such function must be onto. Which one of the following is CORRECT? Correct Answer Only Q and R are true

Concept:

Two functions f and g are inverse of each other if and only if f(g(x)) = g(f(x)) = x

Explanation:

  • Using above property we can say that function f is inverse of itself this means,  f(2)=3 then f(3)=2 but here total elements are 2015 so we can have total 1007 pair but one element is left for some element there exist  f(i) = i. Statement  Q is correct.
  • If function is inverse then function will be onto as well, statement R is also correct.

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