Question

Define a binary relation \(R\) on \(\mathbb{R}\) as \(\\{(x, y) \in \mathbb{R} \times \mathbb{R}: \sin (x)=\sin (y)\\}\). Prove that \(R\) is an equivalence relation. What are its equivalence classes?

Short answer

The equivalence classes are sets of the form \( \{ y : y = x + 2k\pi \} \cup \{ y : y = \pi - x + 2k\pi \} \) for \( x \in \mathbb{R} \).

Step-by-step solution

Defining an Equivalence Relation

An equivalence relation must be reflexive, symmetric, and transitive. We'll need to verify these properties for the relation \( R \) defined by \( (x, y) \in \mathbb{R} \times \mathbb{R} : \sin(x) = \sin(y) \).

Check Reflexivity

To check reflexivity, we need to show that \( (x, x) \in R \) for every \( x \in \mathbb{R} \). Since \( \sin(x) = \sin(x) \), reflexivity holds for relation \( R \).

Check Symmetry

The relation is symmetric if, whenever \( (x, y) \in R \), then \( (y, x) \in R \). If \( \sin(x) = \sin(y) \), it implies \( \sin(y) = \sin(x) \), hence \( R \) is symmetric.

Check Transitivity

The relation is transitive if, whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \). Given \( \sin(x) = \sin(y) \) and \( \sin(y) = \sin(z) \), it follows that \( \sin(x) = \sin(z) \). Thus, \( R \) is transitive.

Concluding Equivalence Relation

Since the relation \( R \) is reflexive, symmetric, and transitive, it is an equivalence relation on \( \mathbb{R} \).

Determine the Equivalence Classes

The equivalence class of a number \( x \) under \( R \) is the set of all real numbers \( y \) such that \( \sin(y) = \sin(x) \). For a given \( x \), \( y = x + 2k\pi \) or \( y = (\pi - x) + 2k\pi \) for some integer \( k \), are all such values of \( y \).

Key concepts

Binary Relations

A binary relation is a concept that involves pairing elements from two sets. These sets can sometimes be the same. In other words, a binary relation on a set can be thought of as a way of showing some kind of relationship or connection between ordered pairs of elements from the set. Specifically, when we mention a binary relation on the set of real numbers, we are referring to a subset of the Cartesian product of the reals, denoted as \(mathbb{R} \times \mathbb{R}\).
For example, consider the relation \(R\) defined by \(\{(x, y) : \sin(x) = \sin(y)\}\). This relation connects any two real numbers \(x\) and \(y\) if their sine values are equal. It's like saying, "x is related to y if they both have the same sine value." Therefore, binary relations can help us understand how different elements, like numbers, are similar or connected through a specific rule or property. Understanding this connection is crucial for recognizing more complex mathematical structures, like functions or equivalence relations.

Reflexivity

Reflexivity is an important property of certain binary relations, which explains whether every element is related to itself. In the context of our relation \(R\), reflexivity means that for every real number \(x\), the pair \((x, x)\) must be in \(R\).
Why is this important? Because relationships where every element relates to itself serve as a foundational part of defining equivalence relations.

• For our specific relation \(R\), since \(\sin(x)\) always equals \(\sin(x)\), the relation includes \((x, x)\) for every real number \(x\).

Thus, the relation \(R\) is reflexive, as all elements meet this criterion. Reflexivity ensures that no element in the set is left out, providing a sense of completeness and internal consistency to the relation.

Symmetry

The next essential element of a binary relation being an equivalence relation is symmetry. This means that if one element is related to another, then the second element is related back to the first. Symmetry in a relation ensures two-way connectivity between elements.
In our context of \(R\), this condition means that if \(\sin(x) = \sin(y)\), then \(\sin(y) = \sin(x)\). Quite simply, if the order in which we pair elements doesn’t matter—like, if \((x, y)\) in the relation guarantees \((y, x)\) is too—then we have a symmetric relation.

• Consider the relation \(R\) as defined by \(\{(x, y) : \sin(x) = \sin(y)\}\). If \((x, y)\) belongs to \(R\), logically \((y, x)\) also belongs to \(R\) because equality is symmetric.

This symmetry is fundamental as it establishes an even level of relationship between elements, creating a balanced connection across the set of elements involved in the relation.

Transitivity

The transitivity of a relation extends the connection created by symmetry. It states that if one element is related to a second, and that second element is related to a third, then the first must also relate to the third. This creates a chain of relationships.
Considering our relation \(R\) on the real numbers, this means that if \((x, y)\) belongs to \(R\) and \((y, z)\) also belongs to \(R\), \((x, z)\) must belong to \(R\). With the definition of \(R\) as \(\sin(x) = \sin(y)\), and \(\sin(y) = \sin(z)\), it directly follows that \(\sin(x) = \sin(z)\).

• This closure across three elements under \(R\) demonstrates that transitivity holds.

Transitivity is essential in maintaining consistent and continuous relationships throughout the set. It allows for the construction of "equivalence classes," where every element related through such a sequence is regarded as one unit or class. Equivalence classes synthesize the property insight gained through reflexivity, symmetry, and transitivity, thus characterizing a well-defined equivalence relation.