3.29. Let S be a finite set. A binary relation on S is simply a collection R of ordered
pairs (x, y) ? S x S. For instance, S might be a set of people, and each such pair
(x, y) ? R might mean "x knows y."
An equivalence relation is a binary relation which satisfies three properties:
• Reflexivity: (x, x) ? R for all x ? S
• Symmetry: if (x, y) ? R then (y, x) ? R
• Transitivity: if (x, y) ? R and (y, z) ? R then (x, z) ? R
For instance, the binary relation "has the same birthday as" is an equivalence
relation, whereas "is the father of" is not, since it violates all three properties.
Show that an equivalence relation partitions set S into disjoint groups
S1, S2,..., Sk (in other words, S = S? U S? U ... U Sk and S? ? S; = Ø for all
i ? j) such that:
• Any two members of a group are related, that is, (x, y) ? R for any
x, y ? Si, for any i.
• Members of different groups are not related, that is, for all i ? j, for all
x ? Si and y ? Sj, we have (x, y) ? R.
(Hint: Represent an equivalence relation by an undirected graph.)
Exercises
