Exercise · ZW-EQUIVALENCE-RELATION-007 · ★★☆ Intermediate
On \(\mathbb{Z}\), let \(a\,R\,b\) if \(a-b\) is even. Is \(R\) an equivalence relation?
Choose one option.
Show answer
- B. Yes, it is an equivalence relation
Explanation
\(a-a=0\) is even; if \(a-b\) is even then \(b-a=-(a-b)\) is even too; and if \(a-b\) and \(b-c\) are even, their sum \(a-c\) is even. The sign does not change parity.
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-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?