MAT 2720 Discrete Mathematics Section 3.3 Relations

Goals Relations Properties of Relations on X

Recall A relation from X to Y is a subset Sometimes, we write Domain of R = all possible value of x Range of R = all possible value of y

Recall A relation from X to X is called a relation on X

Properties of Relation on X R is…If…Diagraph Reflexive Symmetric Transitive

Example 5(a)

Example 5(b)

Example 5(c)

Properties of Relation on X R is…If…Diagraph Antisymmetric(Read)

MAT 2720 Discrete Mathematics Section 3.4 Equivalence Relations

Goals Equivalence Relations A special relation with nice properties. Partition of sets (Clumping Property). Applications to counting problems. CS students should read the applications in p

“Informal” Example Example

“Informal” Example Reflexive?

“Informal” Example Symmetric?

“Informal” Example Transitive?

“Informal” Example “Clumping” Effective

Definitions and Notations is an Equivalence Relation if R is reflexive, symmetric, and transitive.

Example 1 Show that R is an Equivalence Relation

Example 1 Show that R is an Equivalence Relation

Example 1 Proof: ReflexiveAnalysis

Example 1 Proof: SymmetricAnalysis

Example 1 Proof: TransitiveAnalysis

Definitions and Notations is an Equivalence Relation if R is reflexive, symmetric, and transitive. Equivalence Class of :

Example 1

Observations

Partition of a Set (1.1) A partition of a set X is a way to split X into the union of disjoint subsets.

Partition of a Set (1.1) A partition of a set X is a way to split X into the union of disjoint subsets. For every element in X, it belongs to one and only one subset in the partition.

Theorem

“Informal” Example Partition

Theorem

(It is easy to check that R is an equivalence relation.) Example 2

Summary of the 2 Theorems

Theorem