Equivalence Relations


















Example: same-age

Consider the domain of the students in this class.

Suppose a relation is defined on this domain such that
the pair (x,y) is in the relation if x and y are the same age.
(Call this the 'same-age' relation.)

Everyone who is age 22 please stand up.
(You don't need to participate if you'd rather not.)

If there are n people standing, there are n pairs in the relation
for each person who is standing. (a total of n*n pairs for age 22)

Is the 'same-age' relation reflexive? symmetric? transitive?


















What's an Equivalence Relation?

	A relation that is reflexive, symmetric, and transitive.

What key idea is represented by equivalence relations?

	sameness

	same-team
	same-major
	same-size
	same-weight
	same-state


















What's an Equivalence Class?

	A set of objects that are related to each other by an
	equivalence relation.

	Or the set of objects that are equivalent to each other as
	defined by an equivalence relation.

	For example, the set of people who are age 22 is an
	Equivalence Class defined by the 'same-age' relation.


















Partitions

How does the 'same-age' relation split people into groups?

Would there be any person who would be in two different groups?

Would there be any person who would not be in any group?


What's a Partition of a set?

	1. a division of the set into subsets
	2. where each item is found in exactly one subset

Which of X, Y, and Z are partitions of A?

	A = {bob, jim, ann, ned, zed}

	X = {{bob, jim, ann}, {ned, zed}}

	Y = {{bob, jim, ann}, {ann, ned, zed}}

	Z = {{bob, ann}, {ned, zed}}


What's the relationship between Equivalence Relations and Partitions?

	An Equivalence Relation determines a Partition of a set.

	A Partition of a set determines an Equivalence Relation.


















Classwork
You may work with a partner.

For each relation (defined on {1,2,3,4}):

1. State whether the relation is an equivalence relation or not.
2. If it is an equivalence relation, list the equivalence classes.
3. If not, list the properties that are lacking.

	P = {(1,1),(2,2),(3,3),(4,4)}

	R = {(1,1),(1,3),(3,1),(3,3),(3,4),(4,3),(4,4)}

	S = {(1,1),(2,2),(2,3),(3,2),(3,3),(4,4)}