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7. Prove or disprove: Any subset X = {1,2,3,...,2n} with |X|>n contains two (un- equal) elements a, b eX for which alb or bla.

7. Prove or disprove: Any subset X = {1,2,3,...,2n} with |X|>n contains two (un- equal) elements a, b eX for which alb or bla.

Advanced Engineering Mathematics
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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Please answer this question handwritten or typed out in latex. Please do not type out a response that isn't written in latex. Hint: pigeonhole principle.

**Problem 7:** Prove or disprove: Any subset \( X \subseteq \{1, 2, 3, \ldots, 2n\} \) with \( |X| > n \) contains two (unequal) elements \( a, b \in X \) for which \( a \mid b \) or \( b \mid a \). This statement asserts that if you have more than \( n \) elements in a subset of the first \( 2n \) natural numbers, there must be at least one pair of numbers where one divides the other. The task is to either prove this claim or provide a counterexample.
Transcribed Image Text:**Problem 7:** Prove or disprove: Any subset \( X \subseteq \{1, 2, 3, \ldots, 2n\} \) with \( |X| > n \) contains two (unequal) elements \( a, b \in X \) for which \( a \mid b \) or \( b \mid a \). This statement asserts that if you have more than \( n \) elements in a subset of the first \( 2n \) natural numbers, there must be at least one pair of numbers where one divides the other. The task is to either prove this claim or provide a counterexample.
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