Other Relations Related to Transitive Relation

Some of the other types of relations related to the concept of transitive relations are:

• Anti-Transitive Relation
• Intransitive Relation

Let’s discuss these relation in detail.

Anti-Transitive Relation

An anti-transitive relationship works differently. If A is connected with B and B is connected with C then A can’t be connected with C. It’s the opposite of transitivity and is useful in various mathematical contexts.

Example of Anti-Transitive Relation

Let’s define a set of people: {A, B, C, D, E}.

Now, let’s define the “is a parent of” relation:

• A is a parent of B.
• B is a parent of C.

In this relation:

• A is related to B because A is the parent of B.
• B is related to C because B is the parent of C.

However, this relation is anti-transitive because it doesn’t follow the transitive property:

• A is not related to C. A is not the parent of C.

Intransitive Relation

Intransitive relations don’t follow the clear chain rule. If A is connected with B and B is connected with C then it doesn’t guarantee that A is connected with C. Such relations often appear in complex real-world situations.

Example of Intransitive Relation

Consider a relation “feeds on” among animal and their food:

• A feeds on B.
• B feeds on C.

Now, in this relation:

• A is related to B because A feeds on B.
• B is related to C because B feeds on C.

From all this we can’t say for sure that A feeds on C, thus this relation is an example of Intransitive Relation.

Transitive Relation is one of the necessary conditions for equivalence relation, as for any relation to be that needs to to Transitive at first. In Transitive Relation, if element A is related to element B and element B is related to element C, then there must also be a relationship between element A and element C, following the same rule or relation. In other words, if A relates to B and B relates to C, then A must relate to C.

This article provides a well-rounded description of the concept of “Transitive Relation”, including definitions, examples, and properties.