**Problem Statement:** Let $ f $ be a one-to-one function from $ X = \{1, 2, \ldots, n\} $ onto $ X $. Let $ f^k = f \circ f \circ \cdots \circ f $ denote the $ k $-fold composition of $ f $ with itself. Show that there are distinct positive integers $ i $ and $ j $ such that $ f^i(x) = f^j(x) $ for all $ x \in X $. Show that for some positive integer $ k $, $ f^k(x) = x $ for all $ x \in X $.

Problem Statement: Let $ f $ be a one-to-one function from $ X = \{1, 2, \ldots, n\} $ onto $ X $. Let $ f^k = f \circ f \circ \cdots \circ f $ denote the $ k $-fold composition of $ f $ with itself. Show that there are distinct positive integers $ i $ and $ j $ such that $ f^i(x) = f^j(x) $ for all $ x \in X $. Show that for some positive integer $ k $, $ f^k(x) = x $ for all $ x \in X $.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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