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.

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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Question

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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