Question

Suppose \(A \neq \varnothing .\) Since \(\varnothing \subseteq A \times A,\) the set \(R=\varnothing\) is a relation on \(A .\) Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.

Short answer

The relation \(R = \varnothing\) is not reflexive, because no elements exist in the set that would satisfy \((x,x) ∈ R\) for all \(x ∈ A\). However, \(R\) is technically symmetric and transitive, because there are no pairs of elements in \(R\) that contradict these properties.

Step-by-step solution

Reflexivity of the Relation

A relation \(R\) on a set \(A\) is reflexive if for all \(x\) in \(A\) , \((x,x) ∈ R\). But in our case, \(R = \varnothing\), which doesn't have any elements. Therefore, this condition can't be satisfied as there aren't any pair \((x,x)\) within \(R\). Hence, \(R\) is not reflexive.

Symmetry of the Relation

A relation \(R\) on a set \(A\) is symmetric if for all \((x,y)\) in \(A\), \( (x,y) ∈ R \) implies \((y,x) ∈ R\). Our relation \(R\) has no elements, meaning there are no pairs \((x, y)\) in \(R\), so there are no counterexamples to symmetry. So, technically, \(R\) is symmetric.

Transitivity of the Relation

A relation \(R\) on a set \(A\) is transitive if for all \(x, y, z ∈ A\), \((x,y) ∈ R\) and \((y,z) ∈ R\) imply \((x,z) ∈ R\). In this case, since there are no elements in \(R\), there are no pairs \((x, y)\) and \((y, z)\) that would contradict this property. Therfore, \(R\) is transitive.

Key concepts

Reflexivity

When we speak of reflexivity in the context of mathematical relations, we are dealing with a fundamental concept that helps characterize how elements within a set relate to themselves. To put it simply, reflexivity means every element is related to itself. In formal terms, a relation \(R\) on a set \(A\) is considered reflexive if, for every element \((x\) in \(A\), the ordered pair \((x, x)\) is included in the relation \(R\).

This concept builds a foundation for understanding self-connections within a set, which is important in various fields of mathematics such as algebra, geometry, and analysis. A real-world analogy could be a friendship relation, where being friends with oneself would reflect the reflexivity of that relation.

However, in the exercise given, we have a special case where the relation \(R\) is an empty set—there are no elements whatsoever. This lack of elements means that there can be no pair \((x, x)\) in the relation, making it non-reflexive. It's like saying if no one is at a party, no one can be friends with themselves at the party; the required self-connection just isn't there.

Symmetry

Symmetry in mathematical relations corresponds to a type of mirroring or balance. A symmetric relation means if one element is related to a second, then the second is also related to the first—imagine two people who either are both friends with each other, or neither is. More technically, a relation \(R\) on a set \(A\) is symmetric if, whenever \((x,y)\) is in \(R\), then \((y,x)\) is also in \(R\) for all elements \((x,y\) in \(A\).

In our textbook example, the relation \(R\) is empty. This lack of elements means there are no pairs that could possibly violate symmetry—there are no one-way friendships because there are no friendships at all. So paradoxically, the empty relation \(R\) is considered symmetric by default, since there are no counterexamples to the symmetric property.

Transitivity

The concept of transitivity deals with a kind of 'chain reaction' within relationships. If a relation is transitive, and you know that one element is related to a second, and the second to a third, then the relation must also link the first directly to the third. This could be like knowing if person A is taller than person B, and person B is taller than person C, then A must be taller than C. Formally, a relation \(R\) on a set \(A\) is transitive if for any elements \((x,y,z\) in \(A\), whenever \((x,y)\) and \((y,z)\) are in \(R\), then \((x,z)\) must also be in \(R\).

In the exercise, since \(R\) is an empty set and contains no elements, there are no sequences \((x, y\) and \((y, z\) to meet the initial condition of transitivity. Therefore, it does not violate the transitive property and is considered transitive by default. It's a bit like a logic puzzle where no initial clues are given, so all possible solutions must be right.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It is a fundamental part of modern mathematics, underpinning everything from numbers to structures like relations. In set theory, we talk about members (or elements) of sets, subsets, and various operations like union, intersection, and set difference.

In the context of the presented problem, we consider the subsets of Cartesian products, such as \(A \times A\), which represent all possible ordered pairs of the set \(A\)'s elements. When we refer to a relation on \(A\), we're describing a subset of that Cartesian product—a selection of ordered pairs that satisfies certain criteria, like reflexivity, symmetry, and transitivity.

The empty set, denoted \(\varnothing\), has a special place in set theory. It is the set with no elements, and it serves as the identity element for the union operation and absorbs element for the intersection. It's intriguing because while it contains nothing, its very existence allows us to discuss the absence of relationships, as in the example of an empty relation that is symmetric and transitive but not reflexive.