Let R be a relation on a set A. Define a relation ∼ on A as follows: for every a, b ∈ A, a ∼ b if and only if there exist n ∈ N and x0, x1, ..., xn ∈ A such that x0 = a, xn = b and for every k ∈ {1, ..., n}, xkRxk−1 or xk−1Rxk. Show that ∼ is an equivalence relation.

Let R be a relation on a set A. Define a relation ∼ on A as follows: for every a, b ∈ A, a ∼ b if and only if there exist n ∈ N and x0, x1, ..., xn ∈ A such that x0 = a, xn = b and for every k ∈ {1, ..., n}, xkRxk−1 or xk−1Rxk. Show that ∼ is an equivalence relation.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...

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Let R be a relation on a set A. Define a relation ∼ on A as follows: for every a, b ∈ A, a ∼ b
if and only if there exist n ∈ N and x0, x1, ..., xn ∈ A such that x0 = a, xn = b and for every k ∈ {1, ..., n},
xkRxk−1 or xk−1Rxk. Show that ∼ is an equivalence relation.

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