Types of RelationTransitive

While the first element has a relationship with the second element, and the second element has a relationship with the set’s third component, the first element must be related to the third element. For example, if a = b and b = c for three members a, b, and c in set A, then a = c. Equality ‘=’ is a transitive relation in this case. In discrete mathematics, there are primarily three types of relations: reflexive, symmetric, and transitive relations, among others.

Let’s look at the notion of transitive definition, its attributes, and some examples to assist us to grasp the concept better.

What are Transitive Relations?

For a, b, and c in B, transitive relations are well-defined on a set B such that element a must be connected to element c if a is related to b and b is related to c. Let’s look at a case of transitive relations. If and only if a > b, define aRb as a relation R on the set of integers Z. Assume that for integers a, b, and c in Z, aRb and bRc indicate a > b and b > c. We know that for integers, we have a > c whenever a > b and b > c, implying that an is related to c, aRc. As a result, R is a transitive relationship.

Examples of Transitive Relations

Now that we’ve looked at the definition of transitive relations let’s look at some mathematical and non-mathematical examples to help us understand them better.

• The transitive relation ‘is a subset of’ is defined on a power set of sets. A is a subset of C if A is a subset of B and B is a subset of C.

• Let’s consider if A ( a person) is an organic familial of another person B, and B is a biological sibling of C, then A is a biological sibling of C.

• The transitive relation ‘is less than’ is defined on a collection of numbers.

• The transitive relation ‘Is equal to (=)’ is defined on a set of numbers. If a = b and b = c, then a must be equal to c.

• The transitive relation ‘is congruent to’ is defined on the set of triangles. Triangle 1 is congruent to triangle 2, while triangle 2 is congruent to triangle 3.

Definitions Related to Transitive Relations

Let’s look at some examples of transitive relations and their definitions:

• Anti-transitive Relation –

An anti-transitive relation for a, b, and c in A is a binary relation. If a, b, c in A if (a, b) ∈ R and (b, c) ∈ R is always false.

• Intransitive Relation –

If (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R., on a set A, a binary relation R defined is an intransitive relation for some a, b, c in A.

Properties of Transitive Relations

After discussing what is transitive, let us explore some properties of transitive relations:

• A transitive connection’s inverse is also a transitive relation. For example, if ‘is less than is a transitive connection, then the contrary ‘is larger than a transitive relation.

• It is unnecessary for the union of two transitive relations to be transitive. For example, assume R and S are transitive relations, and (x,y) is in R and (y,z) is in S, but (x,z) is not in either.

• A transitive relation is the intersection of two transitive relations. For example, the transitive relations ‘is greater than or equal to’ and ‘is equal to have the intersection connection ‘is equal to,’ which is also a transitive relation.

• If a transitive relation is irreflexive, it is an asymmetric relation.

Important Notes on Transitive Relations

• A transitive relation is defined on an empty set.

• There is no one-size-fits-all formula for calculating the number of transitive relations on a set.

• A transitive relation’s complement does not have to be transitive.

Conclusion

A transitive relation states that if (a,b) and (b,c) are present in the relation set, then (a,c) must also be present. Simply put, if a is linked to b and b is related to c, then a must be related to c. It is known as a transitive definition. When it comes to counterparts, the reflexive connection connects elements from sets A and B in the reverse order, from set B to set A. When the domain and range of two relations are the same, it is called a symmetric relation.