Question

Prove or disprove: If a relation is symmetric and transitive, then it is also reflexive.

Short answer

The statement is false. A counterexample is the relation R = {(1,2), (2,1)} on the set A = {1, 2}, which is symmetric and transitive but not reflexive.

Step-by-step solution

Understand the Problem Statement

This exercise asks to prove or disprove whether a relation that is symmetric and transitive must also be reflexive. That is, if a relation R on a set A has the property that (a,b) in R and (b,a) in R (symmetry) and (a,b) in R and (b,c) in R implies (a,c) in R (transitivity), does it always have the property that (a,a) in R for all a in A (reflexivity)? Analysis of the definitions of these properties is necessary.

Construct a Counterexample

A counterexample is a specific example that refutes a statement. For the statement to be true, it must hold for all possible relations. To disprove this statement, a relation can be constructed that is symmetric and transitive, but not reflexive. Consider the relation R on the set A = {1, 2} where R = {(1,2), (2,1)}. This relation is symmetric, because if (a,b) in R then (b,a) is in R for all a,b in A. The relation is transitive because there are no elements a, b, and c in A such that (a,b) in R and (b,c) in R, but (a,c) not in R. However, it is not reflexive because there are elements a in A such that (a,a) is not in R (in this case, (1,1) and (2,2) are not in R).

Concluding Remarks

The relation R = {(1,2), (2,1)} on the set A = {1, 2} serves as a counterexample that disproves the statement. Thus, it is not necessarily true that a relation that is symmetric and transitive must also be reflexive.

Key concepts

Symmetry in Relations

In the context of mathematical relations, symmetry is a property that imparts a sense of balance. Specifically, if a relation \(R\) on a set \(A\) is symmetric, it means whenever a pair \( (a, b) \) is in \(R\), the pair \( (b, a) \) is also in \(R\). That is, the relationship works both ways: if \(a\) is related to \(b\), then \(b\) is also related to \(a\).

Imagine friends who have a mutual understanding to help each other; if \(a\) helps \(b\), then \(b\) must help \(a\) as well. That's the essence of symmetry in the context of relations. It's important to note, a relation can be symmetric without including every possible pair in the set—what matters is that for every pair that is included, its symmetric counterpart is also present.

Transitivity in Relations

Transitivity is another property that can characterize relationships between elements of a set. For a relation \(R\) on a set \(A\), we say that \(R\) is transitive if whenever we have two pairs \( (a, b) \) and \( (b, c) \) in \(R\), we must also have the pair \( (a, c) \) in \(R\). It's like a chain link of associations—if \(a\) is connected to \(b\), and \(b\) is connected to \(c\), then transitivity demands that \(a\) must be connected to \(c\).

In practical terms, if \(a\) is taller than \(b\), and \(b\) is taller than \(c\), then transitivity indicates that \(a\) is taller than \(c\). This property helps establish order and hierarchy within a set based on the relations defined.

Reflexivity in Relations

Reflexivity in relations is akin to self-acceptance; it relates each element to itself within a set. A relation \(R\) on a set \(A\) is reflexive if each element \(a\) of \(A\) is related to itself—meaning the pair \( (a, a) \) is always in \(R\) for every element \(a\) in \(A\).

Let's use ownership as a simple illustration: if every item belongs to itself, then the relation 'belongs to' is reflexive. It automatically holds that each item is its own proprietor. In a reflexive relation, this principle of self-inclusion must apply universally to all members of the set \(A\).

Counterexamples

Counterexamples are the secret weapon mathematicians use to disprove statements or conjectures. They provide a specific example or case that contradicts a general statement, thus demonstrating its invalidity. In the instance of our initial problem, we disproved the claim that symmetry and transitivity in a relation imply reflexivity by offering a counterexample—an explicit relation that is both symmetric and transitive but fails to be reflexive.

Consider our earlier relation \(R = \{ (1,2), (2,1) \}\) on the set \(A = \{1, 2\}\). It respects symmetry and transitivity, yet it does not include the pairs \( (1,1) \) and \( (2,2) \), which are obligatory for reflexivity. This single counterexample is enough to overturn the assumption that symmetry and transitivity necessitate reflexivity, highlighting the power of counterexamples in mathematical proofs.