Question

Let \(A=\\{a, b, c, d, e\\} .\) Suppose \(R\) is an equivalence relation on \(A .\) Suppose \(R\) has two equivalence classes. Also \(a R d, b R c\) and \(e R d\). Write out \(R\) as a set.

Short answer

The relation \(R\) can be written as the set \(R =\{(a, d), (d, a), (a, e), (e, a), (d, e), (e, d), (b, c), (c, b)\}\)

Step-by-step solution

Identify the equivalence classes

The equivalence class of an element is the set of all elements to which it is related. Given that \(a R d\) and \(e R d\), we can infer that a, d, and e are related, forming one equivalence class. Also given is \(b R c\), which means b and c are related and form the second equivalence class. Therefore, the two equivalence classes are: {a, d, e} and {b, c}.

Form the relation \(R\) as a set

The relation \(R\) can be represented as a set of ordered pairs, where pairs (x, y) are included if x is related to y according to the equivalence relation. From our equivalence classes, we know every element within a class is related to every other element in the same class. Hence for the first class {a, d, e}, we have pairs (a, d), (d, e), (a, e), (d, a), (e, d) and (e, a), and for the second class {b, c}, we have pairs (b, c) and (c, b). Note that because equivalence relations are symmetric, if (a, d) is included in \(R\), then so should be (d, a) and similar for other pairs.

Construct the set \(R\)

\[ R = \{(a, d), (d, a), (a, e), (e, a), (d, e), (e, d), (b, c), (c, b)\} \]

Key concepts

Equivalence Classes

Understanding equivalence classes is essential when dealing with equivalence relations in set theory. An equivalence class groups elements that are considered equivalent under a particular relation. For instance, in our exercise, we identified two equivalence classes within the set A = {a, b, c, d, e} based on the given equivalence relation R. The members of these classes are related to each other, which means they share a certain property defined by R.

By saying that a is related to d, and e is also related to d, we can conclude that a, d, and e are all related to each other, forming one equivalence class. Similarly, b is related to c, hence they form another equivalence class. This notion of grouping helps simplify problems in set theory by clustering elements that have the same relationship to one another. It's like saying that in a school, all students in the same grade are part of an equivalence class since they share the same academic level.

Ordered Pairs

In set theory, the concept of ordered pairs is foundational to understanding relations between elements in sets. An ordered pair, denoted as (x, y), consists of two elements where the first element is x and the second element is y. What makes ordered pairs rather unique is that the order matters—(x, y) is not the same as (y, x) unless x = y.

In our textbook example, we constructed the set R using ordered pairs derived from the equivalence classes. Since the relation R is an equivalence relation, it's reflexive, symmetric, and transitive, necessitating that each element in a class is paired with every other element, including itself. Thus, we included pairs such as (a, d) and (d, a) because equivalence relations are symmetric. Understanding how to create these ordered pairs from the equivalence classes allows students to form a complete representation of the relation R.

Set Theory

Set theory is the mathematical theory of well-defined collections of objects, which are called sets. Within this theory, objects within a set, known as elements, may be anything: numbers, people, letters, etc. In set theory, we use specific notation, such as curly braces {}, to denote sets and simple but powerful operations to combine, relate, and analyze them.

Our textbook problem is a showcase for the application of set theory. We started with a set A and defined an equivalence relation R on it. Our task was to categorize elements into equivalence classes and describe relation R as a set of ordered pairs. Both of these tasks heavily rely on principles from set theory—understanding and manipulating sets. From this, students learn how abstract concepts such as relations and classes can be neatly expressed using the language of mathematics. Set theory forms the foundation for many other fields within mathematics and is hence crucial for students to master.