Question

Write the relation \(<\) on the set \(A=\mathbb{Z}\) as a subset \(R\) of \(\mathbb{Z} \times \mathbb{Z}\). This is an infinite set, so you will have to use set-builder notation.

Short answer

The relation '<' on the set of integers \( \mathbb{Z} \) can be represented as a subset \( R \) of \( \mathbb{Z} \times \mathbb{Z} \) in the set-builder notation as follows: \( R = \{ (a, b) \in \mathbb{Z} \times \mathbb{Z} | a < b \} \).

Step-by-step solution

Understand the Task

We are trying to represent the relation '<' in a more formal mathematical way. This relation consists of ordered pairs of integers from the set of integers, \( \mathbb{Z} \), in which the first integer is less than the second. This set, called \( R \), will be a subset of the Cartesian product \( \mathbb{Z} \times \mathbb{Z} \).

Use Set-Builder Notation

One way to express a subset based on a certain property is by using set-builder notation. In this case, we want to represent all ordered pairs \( (a, b) \) where \( a < b \). Therefore, we can use the set-builder notation as follows: \( R = \{ (a, b) \in \mathbb{Z} \times \mathbb{Z} | a < b \} \).

Key concepts

Mathematical Relations

A mathematical relation is a connection between elements in one set with elements in the same or another set. In our case, we're exploring the relation '<' (less than), which associates each integer in the set of all integers, denoted by \(\mathbb{Z}\), with another integer in the same set according to the rule of one being less than the other.

However, not all relations are as straightforward as the less-than relation. Relations in mathematics can manifest in various forms: they could be reflexive, symmetric, transitive, or a combination of these properties, classifying them as equivalence relations. The understanding of a relation is crucial as it forms the foundation for functions, ordered pairs, Cartesian products, and even more complex structures in set theory and beyond.

To illustrate relations, we use ordered pairs, which brings us to our next concept: ordered pairs and how they play a pivotal role in describing relations.

Ordered Pairs

The ordered pair is a fundamental concept in mathematics, especially when dealing with relations. An ordered pair, written as \( (a, b) \), consists of two elements where order matters - the first element is \( a \) and the second is \( b \). In the context of our relation '<', the ordered pair \( (a, b) \) indicates that \( a \) is related to \( b \) by being less than \( b \).

Essential Properties of Ordered Pairs

Firstly, \( (a, b) \) is different from \( (b, a) \); unless \( a = b \), which in our relation '<', cannot hold true as no integer is less than itself. Secondly, each ordered pair is a distinct element in the Cartesian product of two sets, which leads us into the concept of the Cartesian product and its relevance to defining mathematical relations.

Cartesian Product

The Cartesian product of two sets is the set of all possible ordered pairs obtained by combining each element of the first set with each element of the second set. The notation \( A \times B \) represents the Cartesian product of sets \( A \) and \( B \), where \( A \) is our starting set of elements, and \( B \) is the set we are combining with.

In our exercise, we're focused on the set \(\mathbb{Z} \), the set of all integers. The Cartesian product \(\mathbb{Z} \times \mathbb{Z}\) is the set of all ordered pairs where both elements are integers. From this vast set of ordered pairs, we want to identify a specific subset that defines our relation '<'.

Using Set-Builder Notation

Set-builder notation is the language we use to concisely define this subset. We've defined \( R \) as the set containing all ordered pairs \( (a, b) \) such that \( a < b \) and both \( a \) and \( b \) are part of set \( \mathbb{Z} \). In this notation, \( R = \{ (a, b) \in \mathbb{Z} \times \mathbb{Z} | a < b \} \) succinctly captures the essence of the relation and provides a clear understanding of how the elements are related through the Cartesian product.