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a) Relation R is defined on Z as follows: for every a, b € Z, a R bif 4 | (a - b) (i.e., 4 divides a-b). Is R an equivalence relation? Why or why not? Justify your answer.

a) Relation R is defined on Z as follows: for every a, b € Z, a R bif 4 | (a - b) (i.e., 4 divides a-b). Is R an equivalence relation? Why or why not? Justify your answer.

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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Answer the following questions on equivalence relation and partial order.

**Note:** If you claim that a relation has a specific property which makes it a partial order or an equivalence relation, please explain why you think that the relation has that claimed property. For example, if you claim that a relation is symmetric, you should explain why you think that the relation is symmetric. Follow the same line of argument for all properties. Give examples and/or counterexamples when needed. **a)** Relation \( R \) is defined on \( \mathbb{Z} \) as follows: for every \( a, b \in \mathbb{Z} \), \( a \, R \, b \) if \( 4 \mid (a - b) \) (i.e., 4 divides \( a-b \)). Is \( R \) an equivalence relation? Why or why not? Justify your answer.
Transcribed Image Text:**Note:** If you claim that a relation has a specific property which makes it a partial order or an equivalence relation, please explain why you think that the relation has that claimed property. For example, if you claim that a relation is symmetric, you should explain why you think that the relation is symmetric. Follow the same line of argument for all properties. Give examples and/or counterexamples when needed. **a)** Relation \( R \) is defined on \( \mathbb{Z} \) as follows: for every \( a, b \in \mathbb{Z} \), \( a \, R \, b \) if \( 4 \mid (a - b) \) (i.e., 4 divides \( a-b \)). Is \( R \) an equivalence relation? Why or why not? Justify your answer.
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