A short note on Transitive Relation

In computations, Relation pertains to an established association between two sets of indicators or variables. Sets to be considered as associated with each other should necessarily have elements that are attributed to each other. 

Regarding the properties of relation and functions, if two sets are related to each other, set A has a relation with set B, then the conclusion which can be drawn is that the values of set A are regarded as the domain and the values of set B as the range. There are three categories of relations in mathematics scilicet, reflexive relation, transitive relation, and symmetric. 

What is a transitive relation?

Talking of relation being transitive, this goes for any element in the relation, and not every element of the set should be included in the relation. 

Suppose we take (c, d, e, f) from the given set and (c, d) and (d, e) lie in the relation. Then the relation would be called transitive, if (c, e) is also present in the relation. 

           If (c, d) ∈ R and (d, e) ∈ R 

                 then, (c, e) ∈ R

For example: If we have a set X and it has elements {4,5,6,} and relation as (4,5) and (5,6) which belongs to the set R then the set would be only called transitive when (4,6) is also a part of the relation. 

Unlike the reflexive relation, here we don’t consider all the elements, but any element of the set and if something is excluded it is not necessary to have an ordered pair of that too in the relation. 

Definitions associated with Transitive Relation

1)Intransitive Relation: These are those relations where the first element doesn’t come related to the third element. 

                 If (c, d) ∈ R and (d, e) ∈ R 

                 then, (c, e) ∉ R

2) Anti-transitive Relation: These are those associations where transitivity of relation, (c, d) ∈ R and (d, e) ∈ R does not always indicate that (c, e) ∈ R. 

Properties of Transitive Relation 

Below are some of the distinct properties of a transition relation: 

1)The union of two transitive relations does not necessarily mean the product formed would also be transitively related. However, the intersection of two transitive relations would mean that the outcome of both of them would satisfy the transitive relation. 

2) If a transitive relation is inverted, then the outcome is also a transitive relation. 

3) If a transitive relation does not convince the reflexive relation property then it won’t satisfy the symmetry relation property as well.

Transitive Relation v/s Reflexive Relation and Symmetry Relation

1)Symmetric property

Let us take two elements from a set, say t and u, and say that (t, u) belongs to the relation. If we want our relation to be symmetric, then (u, t) should also belong to the relation. 

              (t, u) ∈ R, then (u, t) ∈ R

For example: If we have a set X and it has elements {4,5,6,}, and the Relation has (4,5) as ordered pairs, then the set would be only called symmetric provided the relation should also have (5,4). Nevertheless, if the relation has (4,6) as pairs but doesn’t have (6,4) in it, then in that case the relation wouldn’t be called a symmetric relation. Unlike the reflexive relation, here we don’t consider all the elements, but any element of the set. If (5,6) is not there in the relation, then it is not necessary that (6,5) should be there in the relation, without that too, the relation can be called a symmetric relation. 

2) Reflexive property

We call any relation reflexive, provided you take an element from the given set X then the ordered pair (g, g) belongs to that relation. This should be for all the elements in that set. 

               g ∈ X, then  (g, g) ∈ R

For example: If we have a set X and it has elements {4,5,6,}, then the set would be only called reflexive when 

                          {4,4} ∈ R  

                          {5,5} ∈ R

                          {6,6} ∈ R

If even one of them doesn’t lie in the relation, then the set cannot be called reflexive. So, if an element says g is related to itself, then it would be called a reflexive relation. 

Transitive relation defines those associations where if the first element is associated with the second one, and the second is related to the third one, the first one should also be certainly related to the third component                 

               If (u, v) ∈ R and (v, w) ∈ R 

                      then, (u, w) ∈ R

Transitive relation example

Example 1: Suppose there is a set Q = {4,5,6,7,8 } that is bestowed by the relation R that states {(r, q): |r-q| is even}, then prove that the above relation mentioned satisfies the Transitive relation. 

If (r, q) ∈ R and (q, s) ∈ R 

                 then, (r, s) ∈ R

Here, in this example, as |r-q| and |q-s| is even, and then |r-s| is also an even number, then the case is transitive. 

Example 2: Biological relation between people is a transitive relation because if Person 1 is related to person 2 and person 2 is related to person 3, then person 1 is also related to person 3.

Example 3: Congruency of the triangle also holds and satisfies the transitive relation. Here, if triangle 1 is congruent to triangle 2 and triangle 2 is congruent to triangle 3, then triangle 1 is congruent to triangle 3 as well. 

Conclusion 

Thus, transitive relation defines those associations where if the first element is associated with the second one, and the second is related to the third one, the first one should also be certainly related to the third component. However, it should also be noted that the union of two transitive relations does not necessarily mean the product formed would also be transitively related. However, the intersection of two transitive relations would mean that the outcome of both of them would satisfy the transitive relation.