1. 1 Relations Rosen 6 th ed., §8.1

2. 2 Relations Re lationships between elements of sets occur in many contextsRe lationships between elements of sets occur in many contexts Example relationships in everyday life: employee and her salary, person and relative, business and it’s phone number, etc.Example relationships in everyday life: employee and her salary, person and relative, business and it’s phone number, etc. In mathematics we study relationships such as those between a positive integer and one that divides it, etc.In mathematics we study relationships such as those between a positive integer and one that divides it, etc.

3. 3 Relations The most direct way to express a relationship between elements of two sets is to use ordered pairs (binary relations)The most direct way to express a relationship between elements of two sets is to use ordered pairs (binary relations) A binary relation from A to B is a set R of ordered pairs where the first element of each ordered pair comes from A and the second element comes from B. We use the notation a R b to denote that (a, b)  R and a a R b to denote that (a, b)  R.A binary relation from A to B is a set R of ordered pairs where the first element of each ordered pair comes from A and the second element comes from B. We use the notation a R b to denote that (a, b)  R and a a R b to denote that (a, b)  R. In mathematics we study relationships such as those between a positive integer and one that divides it, etc.In mathematics we study relationships such as those between a positive integer and one that divides it, etc.

4. 4 Relations Recall the definition of the Cartesian (Cross) Product:Recall the definition of the Cartesian (Cross) Product: –The Cartesian Product of sets A and B, A x B, is the set A x B = { : x  A and y  B}. A relation is just any subset of the CP!!A relation is just any subset of the CP!! –R  AxB Ex: A = students; B = courses.Ex: A = students; B = courses. R = {(a,b) | student a is enrolled in class b}

5. 5 Relations Recall the definition of a function:Recall the definition of a function: f = { : b = f(a), a  A and b  B} Is every function a relation?Is every function a relation? Yes, a function is a special kind of relation Yes, a function is a special kind of relation.Yes, a function is a special kind of relation.

6. 6 Relations on a Set Relations from a set A to itself are of special interestRelations from a set A to itself are of special interest A relation on the set A is a relation from A to AA relation on the set A is a relation from A to A In other words, a relation on a set A is a subset of A x AIn other words, a relation on a set A is a subset of A x A

7. 7 Properties of Relations Reflexivity: A relation R on AxA is reflexive if for all a  A, (a,a)  R.Reflexivity: A relation R on AxA is reflexive if for all a  A, (a,a)  R. Symmetry: A relation R on AxA is symmetric if (x,y)  R implies (x,y)  R.Symmetry: A relation R on AxA is symmetric if (x,y)  R implies (x,y)  R. Anti-symmetry:Anti-symmetry: A relation on AxA is anti-symmetric if (a,b)  R implies (b,a)  R. A relation on AxA is anti-symmetric if (a,b)  R implies (b,a)  R.

8. 8 Properties of Relations Transitivity:Transitivity: A relation on AxA is transitive if (a,b)  R and (b,c)  R imply (a,c)  R. A relation on AxA is transitive if (a,b)  R and (b,c)  R imply (a,c)  R.Examples……..