In studying relation in math, we usually give it a name such as f, or g or h. we the carefully describe the relation and the description is the definition of the relation.

Types of relations

The diagrams below show the different types of relations,

1. One-to-one relations: each object element has just one image. The diagram shows that a maps onto c and b maps onto d.
2. Many-to-one relations: several different objects all have the same image. The diagram shows both a and b maps onto e, and both c and d maps onto f.

In a many-to-one relation, there may be some elements of the domain which map onto more than one range element and some which map onto only one range element.

3. One-to-many relations: one object may have more than one image. In the diagram, a has two images, c and d. in a one-to-many relation some or all the elements of the domain map onto more than one image in the range.
4. Many-to-many relations: one object may have more than one image and also several objects may all map onto the same image. The diagram shows that b maps onto d and f, and also that d is the image of both a and b. in a many-to-many relation, some or all of the range elements are the image of more than one object.

Arrow diagrams

An example of a relation is the relation between a letter of the alphabet ad the letter which follows it. We can call this relation f and define it by writing;

 f : x          y, where y is the letter following x in the English alphabet. In this definition,

• X is called the object
• Y is called the image
• X         y means ‘x maps onto y’

In the mathematical method we can bring out relations of 2 to 8 as shown;

• The relation between 2, 6 and 8 is that, 2+6=8, this relation maps 2 onto 2+6, so we could define this relation as

                        f : x            x + 6

• A relation also exist of 2, 4 and 8, 2×4=8, this relation maps 2 onto 2×4, so we could define this relation as

                        g : x                 4x

• Also 2×5-2= 8, this relation maps 2 onto 2×5-2, so we could define this relation as

                        h : x        5x - 2

• Also 2³=  8, this relation maps 2 onto 2³, so we could define this relation as

                         J : x             x³

Example

Given that the relation which maps x onto x² – 5. The number indicated as w is equal to

Solution

A(w)= (w)² – 5 =-4

                                              W² – 5 = -4

                                               W² = -4+5

                                               W² = 1, w = 1 therefore the relation above is true since w maps onto -4 which also maps onto 1