Can a Relation Be Transitive with Only One Ordered Pair?

The question "Can a relation be transitive with only one ordered pair?" is an intriguing one that blends concepts from set theory and relations. To address this, we need to understand the concept of transitivity in the context of relations on sets.

Understanding Transitivity

A relation R on a set is defined as transitive if and only if whenever ab in R and bc in R, it follows that ac in R. This property ensures a certain logical consistency within the relation.

Example with One Ordered Pair

Consider a relation R with only one ordered pair. For example, let R {(1,2)} be a relation on the set A {1,2,3}. We need to examine if this relation satisfies the transitivity property.

According to the transitivity definition, for R to be transitive, there should not exist any elements x, y, z in A such that xy in R and yz in R, but xz not in R.

Verification

In our example, the only pair in R is (1,2). There are no other pairs in R to form a sequence as required by the transitivity condition. Thus, there are no xy in R and yz in R where xz not in R can be formed. Consequently, the condition for non-transitivity is not met, and R is vacuously transitive.

Therefore, a relation containing only one ordered pair is indeed transitive by default. This property holds because the condition for transitivity cannot be violated since there are no pairs that would require it to be violated.

Generalizing the Concept

The concept can be generalized as follows: if a relation on a set has only one ordered pair, then the relation is transitive. This is because the condition for transitivity (xy in R and yz in R implies xz in R) cannot be violated because there are no such pairs that exist in the relation to begin with.

Conclusion

It is evident that a relation with only one ordered pair is transitive. This is a fundamental concept in set theory and relations, and understanding it can aid in analyzing more complex relations.

By contemplating such instances, we deepen our comprehension of transitivity and solidify our grasp on foundational aspects of mathematical logic and set theory. This understanding is crucial in fields such as computer science, discrete mathematics, and theoretical computer science.