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7. A relation R on a set X is called an equivalence relation if it is symmetric, reflexive, and transitive. (a) Show that congruence modulo m (the relation on Z where we write x =m y if m|(x − y)) is an equivalence relation. (b) Show that every partition P of a set X defines an equivalence relation declaring ~ y if x and y are in the same partition element. by

7. A relation R on a set X is called an equivalence relation if it is symmetric, reflexive, and transitive. (a) Show that congruence modulo m (the relation on Z where we write x =m y if m|(x − y)) is an equivalence relation. (b) Show that every partition P of a set X defines an equivalence relation declaring ~ y if x and y are in the same partition element. by

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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7. A relation \( R \) on a set \( X \) is called an equivalence relation if it is symmetric, reflexive, and transitive. (a) Show that congruence modulo \( m \) (the relation on \( \mathbb{Z} \) where we write \( x \equiv_m y \) if \( m|(x-y) \)) is an equivalence relation. (b) Show that every partition \( P \) of a set \( X \) defines an equivalence relation \( \sim \) by declaring \( x \sim y \) if \( x \) and \( y \) are in the same partition element.
Transcribed Image Text:7. A relation \( R \) on a set \( X \) is called an equivalence relation if it is symmetric, reflexive, and transitive. (a) Show that congruence modulo \( m \) (the relation on \( \mathbb{Z} \) where we write \( x \equiv_m y \) if \( m|(x-y) \)) is an equivalence relation. (b) Show that every partition \( P \) of a set \( X \) defines an equivalence relation \( \sim \) by declaring \( x \sim y \) if \( x \) and \( y \) are in the same partition element.
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