Practice question · Sort into groups
An equivalence relation is reflexive, symmetric AND transitive. Sort each relation.
Groups: Equivalence relation · Not an equivalence relation
- 'equals'
- 'divides'
- 'is less than or equal to'
- 'has the same remainder mod 3 as'
Hints
- All three properties must hold, one failure is enough to disqualify.
- Is 'less than or equal to' symmetric? Is 'divides'?
Show the answer
Equivalence relation: 'has the same remainder mod 3 as', 'equals'
Not an equivalence relation: 'is less than or equal to', 'divides'
Why
Congruence mod 3 and equality satisfy all three properties. 'At most' is reflexive and transitive but not symmetric (2 <= 5 while 5 <= 2 fails), and 'divides' fails symmetry likewise (2 divides 6, but 6 does not divide 2).
Practise Relations and Equivalence
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