Explanation Given information: The relation | and the set N of natural numbers. Proof: Below is the given divisibility: x | y is indicating that x divides by y and some integer k means y = k x . Below is the definition of partially ordered: If relation R defines on a set S , Then partially ordered set is ( S , R ) which is defined by below properties: We can check by Reflexive: if x R x for any x ∈ S . We can check by Transitive: if x R y and y R z then x R z for any x , y , z ∈ S . We can check by Antisymmetry: if x R y and y R x then x = y for any x , y ∈ S . We need to verify the properties of partially ordered set which is the divisibility relation makes the set N as a partially order set. Reflexive: For any a ∈ N , a = | . a . The definition of divisibility a | a for any a ∈ N . Hence a R a for any a ∈ N and divisibility relation | is reflexive. Transitive: Let a R b and b R c it means a | b and b | c for any a , b , c ∈ N . By using definition of divisibility: a | b ⇒ b = k 1 a b | c ⇒ c = k 2 a For some k 1 , k 2 ∈ N Substitute b = k 1 a in c = k 2 b c = k 2 b = k 2 ( k 1 a ) = ( k 2 k 1 ) a = k 3 a ( k 2 k 1 = k 3 ∈ N ) By the definition of divisibility, c = k 3 a implies that a | c

3rd Edition

ISBN: 9781284056242

Author: David J. Hunter

Publisher: Jones & Bartlett Learning

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Chapter 2.5, Problem 1E

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The relation | makes the set N of natural numbers as partially set.

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Chapter 2.4, Problem 40E

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The relation | makes the set N of natural numbers as partially set.

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To show: The relation | makes the set N of natural numbers as partially set.

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Chapter 2 Solutions

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The relation | makes the set N of natural numbers as partially set.