Relations and Functions

1. Relation

Let A and B be two sets. A binary relation from A into B is any subset of the Cartesian product A x B.

Example: 

Let's assume that a person owns three shirts and two pairs of trousers.

So, let A = {yellow shirt, green shirt, red shirt} and

           B = {blue trouser, black trouser}

Then A x B = {(yellow shirt, blue trouser), (green shirt, blue trouser), (red shirt, blue trouser), (yellow shirt, black trouser), (green shirt, black trouser), (red shirt, black trouser)}

• Relation on a Set

A relation from a set A into itself is called a relation on A.

• Properties of Relations

Let's assume that R is a relation on set A; in other words, R ⊆ A x A. Let's write a R b to denote (a,b)∈ R.

1. Reflexive

R is reflexive if for every a ∈ A, aRb.

2. Symmetric

R is symmetric if for every a and b in A, if aRb, then bRa.

3. Transitive

R is transitive if for every a, b and c in A, if aRb and bRc, then aRc.

4. Equivalence

R is equivalence relation on A if R is reflexive, symmetric and transitive.

2. Function

A function, denoted by f, from a set A to a set B is a relation from A to B that satisfies;

- for each element a in A, there is an element b in B such that <a,b> is in a relation, and

- if <a,b> and <a,c> are in the relation, then b=c.

The set A in the above definition is called the domain of the function and B is its co-domain.