2.1 Relation types  (Page 2/3)

Consider a set $A=\left\{\mathrm{1,2,3}\right\}$ . Then, one of the possible reflexive relations can be :

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

However, following is not a reflexive relation :

$$R_1=\{\left(\mathrm{1,1}\right),\left(\mathrm{2,2}\right),\left(\mathrm{1,2}\right),\left(\mathrm{1,3}\right)\}$$

It is not a reflexive relation as one instance of identity relation (3,3) is absent and violates the condition that every element of the set is related to itself.

We state the condition for reflexive relation as :

$$R\text{is reflexive}\iff \left(x,x\right)\in R,\text{for all}x\in A$$

It is clear that identity relation is a reflexive relation. Further, universal relation consists of all combinations of ordered pairs in the Cartesian product. It means it consists of all elements of the identity relation apart from other ordered pairs. Hence, universal relation is also a reflexive relation.

Interpretation of reflexive relation

Reflexivity of a relation (meaning that a relation is reflexive) is used to characterize important algebraic relations. Following relations are reflexive :

• “is equal to”
• “is less than or equal to”
• “is greater than or equal to”
• “divides”
• “is subset of”

The relation “is less than” or “greater than”, however, are not reflexive.

Examples

Problem 1 : Determine whether “greater than or equal to” is a reflexive relation for natural number.

Solution : A relation, “R”, representing “greater than or equal to” is defined as relation on natural number (N) as :

$$\left(x,y\right)\in R\iff x\ge y\text{where}x,y\in N$$

We construct data for “x” and “y” in accordance with the given relation for few initial natural numbers, say 1, 2 and 3, as under :

$$\mathrm{For}x=\mathrm{1,}y=\mathrm{1,2,3}$$

$$Forx=\mathrm{2,}y=\mathrm{2,3}$$

$$Forx=\mathrm{3,}y=3$$

Thus, the relation set is :

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

Evidently, this set consists of relation of all elements of the set, which are related to itself ie. (1,1), (2,2) and (3,3). Thus, we conclude that “is greater than or equal to” is a reflexive relation.

Problem 2 : Determine whether “is not equal to” is a reflexive relation for natural number?

Solution : A relation, “R”, representing “is not equal to” is defined as relation on natural number (N) as :

$$\left(x,y\right)\in R\iff x\ne ywherex,y\in N$$

We construct data for “x” and “y” in accordance with the given relation for few initial natural numbers, say 1,2 and 3, as under :

$$Forx=\mathrm{1,}y=\mathrm{2,3}$$

$$Forx=\mathrm{2,}y=\mathrm{1,3}$$

$$Forx=\mathrm{3,}y=\mathrm{1,2}$$

Thus, the relation set is :

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

Evidently, this set does consists of all ordered pair representing relation of an element with itself. The instances (1,1), (2,2) and (3,3) are missing. Thus, we conclude that “is not equal to" is a irreflexive relation.

Symmetric relation

In symmetric relation, the instance of relation has a mirror image. It means that if (1,3) is an instance, then (3,1) is also an instance in the relation. Clearly, an ordered pair of element with itself like (1,1) or (2,2) are themselves their mirror images. Consider some of the examples of the symmetric relation,

$$R_1=\{\left(\mathrm{1,2}\right),\left(\mathrm{2,1}\right),\left(\mathrm{1,3}\right),\left(\mathrm{3,1}\right)\}$$

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

We have purposely jumbled up ordered pairs to emphasize that order of elements in relation is not important. In order to decide symmetry of a relation, we need to identify mirror pairs. We state the condition of symmetric relation as :