Relation : Type of relations, equivalence relations

Introduction

The relation is one of the critical topics of set theory. Relation, function, and set are interrelated topics. Sets represent the collection of different objectives and relations that define their connection. It is one of the most scoring chapters of mathematics. Hence, in this topic, we will learn about the relations and their categories in detail.

What do you mean by Relations?

The Relations definition is the set of ordered pairs that satisfy a relationship. Let us understand this using two sets. We have two sets, P and Q. Set P has elements 2 and 4 and set Q has elements 10, 20, and 35. We will first take the cartesian product of the two sets. Let’s use an arrow diagram for the cartesian product of sets P and Q.  So, PxQ is a set of the ordered pair as given below:-

SET P={2,4}

SET Q={10,20,35}

(2,10) (2,20) (2,35) (4,10) (4,20) (4,35)

As we can see above, it is two paired with each of these three and then four paired with each of these three. We will get six ordered pairs in the form (p,q) where every p belongs to the set P and every q belongs to the set Q. It can also be written in the set-builder form. Now let’s introduce a relation ‘R’. It gives us a set of all ordered pairs that satisfy its relation. It could be any relation.

Relation ‘R’- All (p,q) satisfying ‘R’.

Now let us introduce another set R’ where, we can say that relation R’ gives us a set of all ordered pairs (p,q) where q is the multiple p.

R’ – q is a multiple of p

R’ = {(p,q):q is a multiple of p,p∈P,q∈Q}

(2,10) (2,20) and (4,20) satisfy relation R’.

R’ = {(2,10),(2,20),(4,20)}

There are three different methods of representing the relations:-

(i) Set Builder – In this form, the relation is defined in a manner that helps find out which all pairs satisfy the given relation. From the above example,

R’ = {(p,q):q is a multiple of p,p∈P,q∈Q} represents the relation in set builder form.

(ii) Roster – In this form, every pair satisfying the relation is written explicitly. From the above example, R’ = {(2,10),(2,20),(4,20)} represents the relation in roster form.

(ii) Arrow Diagram – In this method, the elements of different sets are written in different boxes, and then the elements satisfying the relation are linked through arrows.

Types of Relation in Mathematics

In mathematics, relations are classified into eight (8) types. The different types of relations are briefly elaborated below:-

Empty Relation: In a void or empty relation, there is no relation between any of the set elements. For example, assume set P = {1, 2, 3} then, one of the void relations could be Q = {m, n} where |m – n| = 7. For empty relations, Q = φ ⊂ P × P

Condition: None of the elements of set P will be mapped with other set Q or set P itself. R = ∅ can show the empty relation.

Example: Imagine set P consisting of ten balls in the bin. Then finding a relation R of choosing bats from the bin is impossible. As we know, that basket has only balls and not bats. Hence, the above relation is called a void or an empty relation.

Universal Relation: In a Universal set of relations, every group element is interconnected to each other. For, e.g. Q= R × R

Identity Relation: In Identity Relation, every set element is associated or related to itself only. For, e.g. I = {(a, a), a ∈ A}

Inverse Relation: In Inverse Relation, a set of objects are inversely paired. For, e.g. P-1 = {(b, a): (a, b) ∈ P}

Reflexive Relation: In Reflexive Relation, every object maps itself, and can also map other elements along with it. E.g. : Let set A be (1,2,3) then R = {(1,1), (2,2), (3,3), (1,3)} is an example of reflexive relation in AxA. Therefore, we can also conclude that every identity relation is also a reflexive relation but converse is not true.

Symmetric Relation: In Symmetric Relation, if (p,q) belongs to relation R, then (q,p) must also belong to R.

Transitive relations: These are binary relations defined on a set. If the first element is connected to the second element and the second element is related to the third element, then the first element must be related to the third element. E.g. : if (a,b) and (b,c) belong to relation R then (a,c) must also belong to R.

Equivalence Relation: If the relation is transitive, reflexive and symmetric at the same time then it is considered as an equivalence relation.

What is the difference between Empty Relation, Universal Relation, and Identity Relations?

Conclusion

The relation is one the interesting as well as an essential topic. This topic is a subset of topic sets and functions. In this topic, we have studied the meaning of relations, different types of relations like empty relation, universal relation, and the identity relation. In simple words, the relation means a set of the order of different objects.