Quotient Set of an Equivalence Relation

Asked Feb 04 '19 at 17:25

Active Feb 04 '19 at 17:30

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Let $A = \{1,2,3,4\}$ and $R = \{(1,1) , (1,2) , (2,1) , (2,2) , (3,3), (3,4) , (4,4)\}.$

Is $R$ an equivalence relation on $A$? If so, find its quotient set.

I know that it is an equivalence relation. I found that it was Symmetric, Reflexive, and Transitive. Because it meets these three standards it is an equivalence relation.

But I'm very confused as to what/how I find it's quotient set. Any information is great!

Thanks!

edited Feb 04 '19 at 17:30

cqfd

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asked Feb 04 '19 at 17:25

Grover

Elements in the quotient sets are just equivalence classes under the relation. Can you find the equivalence class of each element in $A$? – Good Morning Captain Feb 04 '19 at 17:28
Possible duplicate of Definition of "quotient set" – Victoria M Feb 04 '19 at 17:29
3

Observe that $(4,3) \notin R$ although $(3,4) \in R$. – little o Feb 04 '19 at 17:29
If you were to include $(4,3)$ in the relation as well... the short answer is that the quotient set is merely the set of equivalence classes. In this case you have the "smaller numbers" and the "larger numbers" and could be written as ${[1],[3]}$ or equivalently as ${{1,2},{3,4}}$ or in several other ways. – JMoravitz Feb 04 '19 at 17:54

Quotient Set of an Equivalence Relation

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