Question

Consider the relation \(R=\\{(x, x): x \in \mathbb{Z}\\}\) on \(\mathbb{Z}\). Is this \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this?

Short answer

The given relation R=\{(x, x): x \in \mathbb{Z}\} on \(\mathbb{Z}\) is reflexive, symmetric and transitive. The relation is known as the identity relation.

Step-by-step solution

Examine Reflexivity

A relation in set A is reflexive when every numbers in the set A is associated with itself. In the given set, all elements x are associated with themselves, thus it is reflexive.

Examine Symmetry

A relation R is symmetric, if for all x and y, if y is related to x then x is related to y. In the given relation R, if (x,x) is there then by transposition (x,x) is also there, therefore, it is symmetric.

Examine Transitivity

A relation R is transitive, if for all x, y and z, if x is related to y and y is related to z, then x is related to z. In the set, if (x,x) and (y,y) both are there then (x,y) should also be there for it to be transitive. For it to be true, x and y must be the same. If they are the same, then (x,y) is also part of the set, meaning the set is transitive.

Identifying the Relation

The given relation is the identity relation over the integers. Identity relation is a relation which relates every element of a set to itself only, exactly what our R does here.

Key concepts

Reflexive Relation

In the realm of set theory and mathematics, a reflexive relation is a foundation block for understanding how elements within a set can relate to themselves. Think of it as a kind gesture where every member in a group is mirroring an action back to itself—like giving oneself a high-five! For a relation to be considered reflexive over a set, it must simply satisfy one condition: every single element within that set must be paired up with itself.

Visualize a set of numbers, where each number has a loop drawn from it back to itself. This neatly aligns with the relation described in the exercise, where the relation, denoted as \(R\), consists of pairs of integers in the form of \((x, x)\). This means every integer \(x\) from the set of integers \(\mathbb{Z}\) is 'high-fiving' itself, fulfilling the condition needed to make \(R\) a reflexive relation.

Symmetric Relation

Symmetry in mathematics conveys a sense of harmony and balance. When we examine symmetric relations, think of a dance duo gracefully mirroring each other's steps. Specifically, a relation is symmetric if whenever one element is related to a second, the second is related to the first in return. It’s like an equal exchange of communication between friends.

In our exercise, the relation \(R\) displays this symmetry. The relationship established is like an echo; for every pair \((x, x)\), its mirror image would be \((x, x)\) as well—no more, no less—as every number is coupled with itself. This characteristic reflects perfect symmetry within the relation. Although it might seem trivial, it exemplifies symmetry since we can't find a single pair that breaks this echo-like condition.

Transitive Relation

To understand transitive relations, envisage a relay race where the baton is smoothly handed over from person to person. In such relations, if the first person hands off the baton to the second, and the second to the third, then the first should be able to hand it to the third directly. If a relation consistently does this with its elements, it's then deemed transitive.

The exercise presents us with a relation \(R\) which possesses this transitive nature. Because each element is related to itself, and there exists no case where a third distinct element would disrupt this chain, the relation continuously maintains the 'baton passing' from every element back to itself, upholding the essence of transitivity.

Relation Properties

Diving deeper into the characteristics of relations, we will explore some fundamental properties that define the ways in which elements can be related within a set. The properties—including reflexive, symmetric, and transitive—are akin to personality traits of a relation, setting the ground rules for how the elements interact.

Our exercise touches on a special type of relation termed the identity relation. Such a relation is a prime example of all these properties at their most basic level, tying every element in a set exclusively to itself, like a tailored fit for each member. With the identity relation, there are no surprises or complex inter-member relationships—it’s the mathematical equivalent of self-acknowledgement and self-sufficiency.