Abstract Algebra: Theory and Applications · Problem 1.18

Preliminaries — Thomas Judson

Mathematics — Abstract Algebra · equivalence-relations

Problem

Which of the following are equivalence relations on

\mathbb{Z}

? For those that are equivalence relations, describe the equivalence classes. ((a))

x \sim y

x \leq y

; ((b))

x \sim y

xy > 0

; ((c))

x \sim y

x = y

x = -y

; ((d))

x \sim y

x - y

is divisible by

n

, where

n \in \mathbb{N}