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### Relations

Rosen 6th ed., §8.1

Relations

- Relationships 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.
- In mathematics we study relationships such as those between a positive integer and one that divides it, etc.

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 aRb to denote that (a, b) ∉ R.
- In mathematics we study relationships such as those between a positive integer and one that divides it, etc.

Relations

- 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!!
- R⊆ AxB
- Ex: A = students; B = courses.

R = {(a,b) | student a is enrolled in class b}

Relations

- Recall the definition of a function:

f = { : b = f(a) , aA and bB}

- Is every function a relation?

Yes, a function is a special kind of relation

Relations on a Set

- Relations from a set A to itself are of special interest
- A 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 A

Properties of Relations

- 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 (a,b) ∈ R implies (b,a) ∈ R for all a,b ∈A.
- Anti-symmetry:

A relation on AxA is anti-symmetric for all a,b ∈A, if (a,b) ∈ R and (b,a) ∈ R, then a=b.

Properties of Relations

- Transitivity:

A relation on AxA is transitive if (a,b) R and (b,c) R imply (a,c) R.

Examples……..

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