2.1 Relation types

Relations among elements of a set have wide possibilities. A systematic approach to study them is facilitated by recognizing different relation types. It should be noted that all relation types described here are relation on one set.

We describe a relation on set itself as :

Relation on A

A relation “R” from set “A” to “A” is called a “relation on A”.

In this module, we shall be using a symbol, “xRy” to denote an instance of relation (ordered pair). The symbol conveys that the instance of relation denoted by the symbol is an ordered pair (x,y), which follows relation “R”.

Void relation

Relation is a subset of Cartesian product of two sets. We have seen that power set of Cartesian product “ $A\times B$ ” is a set of all possible relations among the elements of sets “A” and “B”. In the case of “relation on A”, the power set of Cartesian product “ $A\times A$ ” is a set of all possible relations among the elements of set “A”.

One of the subsets of the power set is empty set or void set. This subset without any element is called the void relation.

$$R=\phi =\left\{\right\}$$

Universal relation

Universal relation is the widest possible relation. This relation consists of all ordered pairs of the Cartesian product “ $A\times A$ ”.

$$R=A\times A$$

Consider a set $A=\left\{\mathrm{1,2,3}\right\}$ . Then, universal relation set is :

$$R=\{\left(\mathrm{1,1}\right),\left(\mathrm{1,2}\right),\left(\mathrm{1,3}\right),\left(\mathrm{2,1}\right),$$

$$\left(\mathrm{2,2}\right),\left(\mathrm{2,3}\right),\left(\mathrm{3,1}\right),\left(\mathrm{3,2}\right),\left(\mathrm{3,3}\right)\}$$

Identity relation

An identity relation is defined as :

Identity relation

In an identity relation "R", every element of the set “A” is related to itself only.

Note the conditions conveyed through words “every” and “only”. The word “every” conveys that identity relation consists of ordered pairs of element with itself - all of them. The word “only” conveys that this relation does not consist of any other combination.

Consider a set $A=\left\{\mathrm{1,2,3}\right\}$ . Then, its identity relation is :

$$R=\{\left(\mathrm{1,1}\right),\left(\mathrm{2,2}\right),\left(\mathrm{3,3}\right)\}$$

It is evident that a set has only one such relation. This relation, as we can see, identifies the set - as it identifies each elements of the set, which are related to itself. By looking at the relation, we can identify the set itself. For this reason, the name of this relation is identity relation. In set builder form, we express an identity relation as

$$R=\{\left(x,x\right):\text{for all}x\in A\}$$

The qualification of the relation is that first and second element of the ordered pair is same element, which belongs to set A.

The followings are not an identity relation :

$$R_1=\{\left(\mathrm{1,1}\right),\left(\mathrm{2,2}\right)\}$$

$$R_2=\{\left(\mathrm{1,1}\right),\left(\mathrm{2,2}\right),\left(\mathrm{3,3}\right),\left(\mathrm{1,2}\right),\left(\mathrm{1,3}\right)\}$$

First one is not an identity relation as it does not include the pairing of remaining element “3”. Second is not an identity relation, because there are other combinations of pairs in the relation.

Reflexive relation

Reflexive relation is an expansion of identity relation. In the simple word, reflexive relation is plus identity relation.

Reflexive relation

In reflexive relation, "R", every element of the set “A” is related to itself.

The definition of reflexive relation is exactly same as that of identity relation except that it misses the word “only” in the end of the sentence. The implication is that this relation includes identity relation and permits other combination of paired elements as well.