Exercise · ZW-EQUIVALENCE-CLASS-010 · ★★★ Advanced
If \([a]\cap[b]\neq\varnothing\), what holds?
Choose one option.
Show answer
- A. \([a]=[b]\)
Explanation
If \(y\in[a]\cap[b]\), then \(y\,R\,a\) and \(y\,R\,b\). By symmetry and transitivity \(a\,R\,b\), so \([a]=[b]\). It does not follow that \(a=b\): on \(\{1,2,3\}\) with parity, \(1\) and \(3\) share a class without being equal.
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-007 On ℤ, let a R b if a and b leave the same remainder when divided by 3. How many equivalence classes…
- 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]?