Equivalence relations - Math Central

HOME

Math Central

Quandaries & Queries

Question from jim, a teacher:

Determine whether the given relation is an equivalence relation on the
set. Describe the partition arising from each equivalence relations.
1. x is Related to y in Real number if |x| = |y|.
2. n is Related to m in Z+ if n and m have the same number of digits in the usual base ten notation.

Let n be a particular integer in Z+. Show that congruence modulo n is an equivalence relation on Z.

Hi Jim,

A relation on a set S is an an equivalence relation if it satisfies three properties

If x is in S the x is related to x. (reflexive property)
If x and y are in S and x is related to y then y is related to x. (symmetric property)
If x, y and z are in S and x is related to y and y is related to z then x is related to z. (transitive property)

Your first relation is reflexive (|x| = |x|) and symmetric (if |x| = |y| then |y| = |x|) and transitive (if |x| = |y| and |y| = |z| then |x| = |z|) so the relation is an equivalence relation.

Now try the second relation. Does it satisfy all three properties?

For the third relation use the properties of congruence to show it is an equivalence relation.

Harley

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.