Exploring the Identity Relation and Its Transitivity

In the realm of mathematics and logic, understanding different types of relations is crucial. One such important relation is the identity relation, and an intriguing property of this relation is that it is also a transitive relation. This article aims to explore the definitions of both these concepts and delve into the proof that the identity relation is indeed transitive.

Understanding Identity Relation

The concept of an identity relation is rooted in set theory and formal logic. It refers to the set of ordered pairs where each element in a given set A is related to itself. Mathematically, the identity relation on a set A can be denoted as:

I {(a, a) | a ∈ A}

To put it simply, every element in set A is paired with itself. This means that for any element a in set A, the ordered pair (a, a) is included in the identity relation I.

Understanding Transitive Relation

A relation R on a set A is considered transitive if whenever (a, b) ∈ R and (b, c) ∈ R, it follows that (a, c) ∈ R for any a, b, c in A. In simpler terms, if an element a is related to b, and b is related to c, then a must also be related to c.

Proving the Transitivity of Identity Relation

To prove that the identity relation is transitive, we will use the definitions of both the identity relation and the transitive relation. Let's break down the proof step by step:

Assume (a, b) ∈ I and (b, c) ∈ I.

By the definition of the identity relation, if (a, b) ∈ I, then a b.

Similarly, if (b, c) ∈ I, then b c.

From these equalities, we can conclude the following:

Since a b and b c, it follows that a c.

The pair (a, c) can be expressed as (a, a) because in the identity relation, the only pairs are of the form (x, x).

Therefore, (a, c) is not in the identity relation unless a c.

Thus, we can state that since (a, c) is not in the identity relation unless a c, it follows that (a, c) ∈ I if (a, b) ∈ I and (b, c) ∈ I. This confirms that the identity relation satisfies the condition for transitivity.

Summary

The identity relation is transitive because it only relates elements to themselves. When (a, b) and (b, c) are both in the identity relation, it necessarily implies a b and b c, leading to a c, thus confirming that (a, c) is also part of the identity relation.

Non-trivial Example: Modulus 5 Arithmetic

Consider the arithmetic in mod 5, also known as modulus 5 arithmetic. In this setup, we consider all numbers that have the same remainder when divided by 5 as equal. For example, 1, 6, and 11 are considered identical because they all have the same remainder of 1 when divided by 5. Similarly, 2, 7, and 17 are considered identical because they each have a remainder of 2 when divided by 5. To verify that this relation is transitive, we can check any triple (a, b, c) in this setup:

If (1, 6) and (6, 11) are in the relation, then 1 ≡ 6 (mod 5) and 6 ≡ 11 (mod 5). Therefore, 1 ≡ 11 (mod 5), which means (1, 11) is also in the relation.

To generalize, if a ≡ b (mod 5) and b ≡ c (mod 5), then a ≡ c (mod 5). This confirms that the relation is transitive.

This example shows that the identity relation in this context is also transitive, further emphasizing the mathematical principles we discussed earlier.

Conclusion

In conclusion, the identity relation and transitivity are interrelated concepts in mathematics. The identity relation, defined as a relation where each element is related to itself, is inherently transitive due to the reflexive and symmetric properties. This article has provided a detailed explanation of both concepts and a proof of the transitivity of the identity relation, along with an illustration through modulus 5 arithmetic.