Week 3 - Day 2: Equivalence Relations and Rational Numbers

This session focuses on various examples of equivalence relations, including visual representations, applications to geometry, and algebraic structures. We also cover the basic theory behind equivalence relations, equivalence classes, and how they relate to the construction of rational numbers.

Examples of Equivalence Relations

We begin with some visual and geometric examples of equivalence relations:

• Cylinder: The equivalence class is represented by the set \([a] = \{a, b\} = [b]\), showing that points \(a\) and \(b\) are equivalent on the cylindrical surface.
• Möbius Strip: In this case, \([a] \sim [b]\) introduces a twist, demonstrating how the Möbius strip changes the nature of the equivalence relation.
• Torus: A torus is created by joining opposite edges of a rectangle to form a cylinder, followed by connecting the top and bottom edges of the cylinder. This surface has no boundaries, and equivalence classes can be visualized similarly.
• Klein Bottle: The Klein bottle is a non-orientable surface similar to the Möbius strip, but it requires four-dimensional space to exist without intersection.

Additional Examples of Equivalence Relations

• Similar Triangles: Triangles are equivalent if their corresponding angles are equal, and their sides are proportional. For example, \(\frac{5}{4} \sim \frac{10}{8}\) demonstrates that triangles with proportional sides are equivalent.
• Coterminal Angles: Two angles are coterminal if they differ by a full rotation (i.e., multiples of \(360^\circ\)). For example, \(10^\circ \sim 370^\circ\).
• Knots: Knots are equivalent if one can be deformed into the other without cutting or passing through itself. This is an important concept in knot theory and topology.

Basic Theory of Equivalence Relations

Now, let’s review the formal definitions and properties of equivalence relations:

Definition of Equivalence Relation

A relation \(\sim\) on a set \(S\) is called an equivalence relation if it satisfies the following three properties:

• Reflexive: For all \(a \in S\), \(a \sim a\).
• Symmetric: For all \(a, b \in S\), if \(a \sim b\), then \(b \sim a\).
• Transitive: For all \(a, b, c \in S\), if \(a \sim b\) and \(b \sim c\), then \(a \sim c\).

Equivalence Classes

Suppose \(\sim\) is an equivalence relation on a set \(S\). Given an element \(a \in S\), the equivalence class of \(a\) is the set of all elements \(b \in S\) that are equivalent to \(a\). This is denoted as:

Theorem: Equivalence of Classes

If \(\sim\) is an equivalence relation on a set \(S\), then for any \(a, b \in S\), we have:

This means that two elements are equivalent if and only if they belong to the same equivalence class.

Equivalence Classes and the Set \(\bar{S}\)

We can collect all the equivalence classes into a set, denoted by:

This set is smaller than \(S\), because elements that are equivalent get "glued together" into a single class.

Function \(\pi: S \to \bar{S}\)

We define a function \(\pi\) from \(S\) to \(\bar{S}\), where \(\pi(x)\) is the equivalence class of \(x\). In symbols:

This function is surjective, meaning that for every equivalence class in \(\bar{S}\), there is an element in \(S\) that maps to it.

Equivalence Classes in \(S\)

In the set \(S\), we have the implication:

In the quotient set \(\bar{S}\), we simplify this to:

Example: Pairs of Integers

Let’s define \( S = \mathbb{Z} \times \mathbb{N} \), the set of ordered pairs where the first element comes from the integers and the second from the natural numbers:

Defining an Equivalence Relation on \(S\)

We define the equivalence relation \(\sim\) on \(S\) by the following rule:

This relation captures a key idea that the cross-product of the two pairs must be equal.

Examples of the Relation

• \((2, 3) \sim (2, 3)\) because \(2 \cdot 3 - 3 \cdot 2 = 0\). \(\checkmark\)
• \((2, 3) \not\sim (6, 1)\) because \(2 \cdot 1 - 6 \cdot 3 = -16\). \(\times\)
• \((2, 3) \sim (4, 6)\) because \(2 \cdot 6 - 3 \cdot 4 = 0\). \(\checkmark\)

Equivalence Class of \((2, 3)\)

We can now identify the equivalence class of \((2, 3)\):

This set contains all pairs that are equivalent to \((2, 3)\) under our relation \(\sim\).

Defining Rational Numbers

We can now introduce a definition based on these equivalence classes. The rational number \(\frac{2}{3}\) can be understood as the equivalence class of the pair \((2, 3)\):

The Set of Rational Numbers \(\mathbb{Q}\)

We can generalize this construction to define the set of rational numbers \(\mathbb{Q}\) as the set of all equivalence classes of pairs \((a, b)\) where \(a \in \mathbb{Z}\) and \(b \in \mathbb{N}\) (with \(b \neq 0\)):