Example of Transitive Relation

Example 1: Imagine a set of students and define a relation “is taller than” as follows:

• Alice is taller than Bob.
• Bob is taller than Carol.

We want to know if this relation is transitive.

Solution:

To check for transitivity, we ensure that if A is connected with B and B is connected with C then A must also be connected with C.

Given our relation:

• Alice is taller than Bob.
• Bob is taller than Carol.

According to the transitive property, since Alice is taller than Bob and Bob is taller than Carol, it must be the case that Alice is also taller than Carol for the relation to be transitive.

In this case, Alice is indeed taller than Carol and the is taller than relation is transitive among these students.

Example 2: Let’s consider a set of numbers and define a relation “is divisible by” as follows:

• 12 is divisible by 3.
• 3 is divisible by 1.

We want to determine if this relation is transitive.

Solution:

To check for transitivity, we need to ensure that if A is divisible by B and B is divisible by C, then A must also be divisible by C for the relation to be transitive.

Given our relation:

• 12 is divisible by 3.
• 3 is divisible by 1.

According to the transitive property, since 12 is divisible by 3 and 3 is divisible by 1, it must be the case that 12 is also divisible by 1 for the relation to be transitive.

In this case, 12 is indeed divisible by 1 (12 divided by 1 equals 12), and the is divisible by relation is transitive among these numbers.

Example 3: Let’s consider a group of animals and define a relation “is a predator of” as follows:

• Animal X is a predator of Animal Y.
• Animal Y is a predator of Animal Z.

We want to determine if this relation is transitive.

Solution:

To check for transitivity, we need to ensure that if Animal X is a predator of Animal Y and Animal Y is a predator of Animal Z, then it must be the case that Animal X is also a predator of Animal Z.

Given our relation:

• Animal X is a predator of Animal Y.
• Animal Y is a predator of Animal Z.

According to the transitive property, since Animal X is a predator of Animal Y and Animal Y is a predator of Animal Z, it logically follows that Animal X must be a predator of Animal Z for the relation to be transitive.

In this case, if Animal X is a predator of Animal Y and Animal Y is a predator of Animal Z, it indeed means that Animal X is also a predator of Animal Z. Therefore, the “is a predator of” relation is transitive among these animals.

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.