17.5.2. Examples of Transitive Relations

A relation R is called a transitive relation if the following universal quantification is true. We want that if at all a R b and b R c in your relation, then a also should be related to c.

A transitive relation indicates a logical connection among three elements of a set. Specifically, if 'a' is related to 'b' and 'b' is related to 'c', then it must follow that 'a' is also directly related to 'c'. This property creates a chain-like relationship, making it essential for reasoning and deduction in mathematics and logic.

Imagine a family tree where if person A is a parent of person B, and person B is a parent of person C, then naturally, person A is a grandparent of person C. This connection neatly illustrates the concept of transitivity where the relationship can be inferred across multiple generations.

In terms of graph theoretic properties, if you have an edge from a to b in the graph of your relation R. And if you have a directed edge from the node b to the node c in the graph of your relation R. Then we need that there should be an edge from a to c as well.

In graph theory, transitive relations can be visualized using directed graphs where nodes represent elements and edges represent relations. If an edge (arrow) connects nodes 'a' to 'b' and 'b' to 'c', a transitive relation necessitates the presence of a direct edge from 'a' to 'c'. This visual representation helps understand how relations can be extended through transitivity in a systematic way.

Think of a transit system: if a bus travels from Station A to Station B, and then another bus connects Station B to Station C, passengers can effectively travel from Station A to Station C through these connections, demonstrating transitivity in transportation routes.

Consider the first relation it is a transitive relation, of course, so here everything is defined over a set say {1, 2} and a relation R is transitive. Because you have; (1, 1) present which can be also considered as (a, b) as well as (b, c) as well as (a, c). So, again the same is true for (2, 2). But your relation R is not transitive because you have a case here where you have (a, b) present, you also have (b, c) present but no (a, c) is present here.

In this specific case, the relation defined with elements from the set {1, 2} illustrates transitivity. When evaluating the pairs, if (1, 1) is included, it affirms that 1 is related to itself. This can extend to combinations with other relations, like (1, 2) and (2, 1). However, if we observe a situation where (1, 2) and (2, 3) are present, but (1, 3) is not, that violates transitivity as it breaks the necessary condition.

This can be likened to a game of telephone. If player A tells player B a secret (A -> B) and player B tells player C the same secret (B -> C), then in a transitive scenario, player A should also have a direct way to pass the secret to player C (A -> C). If they cannot connect directly, then the transitive property fails in this game.

Your relation R is also a transitive relation because it vacuously satisfies this implication because at the first place there is no (a, b) and (b, c) present in your R.

In scenarios where certain pairs are absent from the relation, we say the condition is vacuously satisfied. For instance, if there are no relations (like (a, b) or (b, c)), the requirement for transitivity is inherently met since there is nothing to contradict it. This might seem counterintuitive, but logically it holds true since the implication does not fail due to absence.

Consider a classroom where no student has ever arrived late. In this situation, if we consider students getting from home to school by way of each other, it doesn't contradict the idea of being on time if no late students exist in the first place. Thus, the classroom's punctuality can be said to vacuously satisfy the rule for timely arrivals.

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