Exercise · ZW-EQUIVALENCE-RELATION-009 · ★★☆ Intermediate
On \(A=\{a,b,c\}\), which of these relations is an equivalence relation?
Choose one option.
Show answer
- D. \(\{(a,a),(b,b),(c,c)\}\)
Explanation
Only the diagonal satisfies all three properties. The sets containing \((a,b)\) without \((b,a)\) fail symmetry, and the one lacking \((c,c)\) fails reflexivity.
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-006 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?
- 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?