Let f be a mapping from x = {1, 2, 3, ......., 50} to itself such that for m, n
x, m - n implies that f(m) - f(n). then which of the following is true

Rucha rajesh shingvekar

Concept: If a mapping of a function is defined to itself, it means the range and domain of the function are the same. Calculation: In this problem nothing has been mentioned about the property or nature of function so, we will proceed with the option elimination method. Given: X = {1, 2, 3, ....., 50}, and for m, n ϵ X, m ≤ n implies that f(m) ≤ f(n) Based on the inequality mentioned we can conclude that, From the subset of X, we can assign two variables m, n such that m ≤ n. For every m ≤ n, there must be a function f(m) ≤ f(n). For m=1, since f(m) cannot be any less than 1, so f(1) = 1, similarly For m=2,  f(m) can be m, or m - 1, so f(2) = 1 or 2, and so on Comming to options, Option (1) There is m ϵ X such that f(m) = m It is correct since, for m = 1, we must have f(1) = 1, otherwise, the inequality will not hold true. Option (2) For every m ϵ X, we may have f(m) = m - 1 It is incorrect since, for m = 1, we must have f(1) = 1 For m = 2, f(m) can be 1 or 2 For m = 3, f(m) can be 1 or 2 or 3 So it cannot be true for every m ϵ X Option (3) For every m ϵ X, we may have f(m) = m + 1 It is incorrect since, for m = 1, we must have f(1) = 1 For m = 2, f(m) can be 1 or 2 For m = 3, f(m) can be 1 or 2 or 3 So it can never be true for every m ϵ X Option (4) For every even m ϵ X, we must have f(m) = 1/2 m Since nothing has been mentioned about the nature of the function or the function itself, we cannot assure it for every even m ϵ X. Hence, we can say that option (1) is correct.