Exercise · ZW-EQUIVALENCE-CLASS-007 · ★★☆ Intermediate
On \(\mathbb{Z}\), let \(a\,R\,b\) if \(a\) and \(b\) leave the same remainder when divided by 3. How many equivalence classes are there?
Choose one option.
Show answer
- D. 3
Explanation
The possible remainders when dividing by 3 are 0, 1 and 2, and each defines one class. There are infinitely many integers, but only three classes.
Review the concept: Equivalence class
Given an equivalence relation \(R\) on \(A\) and an element \(a\in A\), the equivalence class \([a]\) is the set of all \(x\in A\) with \(x\,R\,a\).
\[[a]=\{x\in A : x\,R\,a\}\]
More practice problems: equivalence class
- ZW-EQUIVALENCE-CLASS-001 What is the equivalence class [a]?
- ZW-EQUIVALENCE-CLASS-002 What is the quotient set A/R?
- ZW-EQUIVALENCE-CLASS-003 Does a belong to its own class [a]?
- ZW-EQUIVALENCE-CLASS-004 On \1,2,3,4,5,6\ with the relation «same parity», how many equivalence classes are there?
- ZW-EQUIVALENCE-CLASS-005 If x∈[a], what holds?
- ZW-EQUIVALENCE-CLASS-006 On ℤ, let a R b if a-b is a multiple of 3. What is [0]?
- ZW-EQUIVALENCE-CLASS-008 Let A=\1,2,3,4\ and R an equivalence relation with classes \1,2\ and \3,4\. What is [2]?
- ZW-EQUIVALENCE-CLASS-009 On A=\a,b,c\, the equivalence relation is equality. How many classes are there?