Exercise · ZW-EQUIVALENCE-RELATION-006 · ★★☆ Intermediate
Let \(A=\{1,2,3,4\}\) and \(R=\{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)\}\). Is \(R\) an equivalence relation?
Choose one option.
Show answer
- C. Yes: it is reflexive, symmetric and transitive
Explanation
Every element has its own reflexive pair, \((1,2)\) and \((2,1)\) appear together, and the only two-step chains, \(1\to2\to1\) and \(2\to1\to2\), close on pairs already present. Relating 3 to 4 is not a requirement.
Review the concept: Equivalence relation
A binary relation \(R\) on \(A\) that is reflexive, symmetric and transitive. It groups the elements of \(A\) that behave alike with respect to some property.
\[\forall a\in A\,(a\,R\,a)\ \wedge\ \forall a,b\in A\,(aRb \Rightarrow bRa)\ \wedge\\ \forall a,b,c\in A\,(aRb\wedge bRc\Rightarrow aRc)\]
More practice problems: equivalence relation
- ZW-EQUIVALENCE-RELATION-001 Which three properties must a relation R on A satisfy to be an equivalence relation?
- ZW-EQUIVALENCE-RELATION-002 What does it mean for R to be symmetric?
- ZW-EQUIVALENCE-RELATION-003 What does reflexivity require?
- ZW-EQUIVALENCE-RELATION-004 What does transitivity require?
- ZW-EQUIVALENCE-RELATION-005 Is equality = on a set A an equivalence relation?
- ZW-EQUIVALENCE-RELATION-007 On ℤ, let a R b if a-b is even. Is R an equivalence relation?
- ZW-EQUIVALENCE-RELATION-008 On ℤ, let a R b if a≤ b. Is R an equivalence relation?
- ZW-EQUIVALENCE-RELATION-009 On A=\a,b,c\, which of these relations is an equivalence relation?