Sets: Fundamentals and Representation

Introduction to Sets: Definition and Basic Concepts

In mathematics, a set is one of the most fundamental and basic concepts. It provides a framework for grouping objects together. These objects, which can be anything from numbers, letters, symbols, points, or even other sets, are called the elements or members of the set.

Definition of a Set

A set is formally defined as a well-defined collection of distinct objects.

• Well-defined Collection:

  This is a crucial aspect of the definition. A collection is considered well-defined if there is an absolute and clear criterion for determining whether any given object belongs to the collection or not. There should be no ambiguity, opinion, or subjectivity involved in deciding membership.

  Illustration:

  Consider the following collections:

  • "The collection of all rivers in India." - This is a well-defined collection. We can clearly list all rivers in India and definitively say whether a body of water is an Indian river or not. Hence, it forms a set.
  • "The collection of brilliant students in a class." - This is not a well-defined collection. The term "brilliant" is subjective and varies from person to person. There is no objective criterion to decide which students are "brilliant" and which are not. Hence, it does not form a set.
  • "The collection of months of a year beginning with the letter 'J'." - This is a well-defined collection. The months are January, June, and July. For any given month, we can definitively say if it starts with 'J' or not. Hence, it forms a set.

• Distinct Objects:

  The objects within a set must be unique. Each element is considered only once, regardless of how many times it might appear in a descriptive list or collection. Repetition of elements within the set notation is ignored. For example, the collection of letters in the word "GOOGLE" contains the letters G, O, O, G, L, E. As a set, we list only the distinct letters: $\{G, O, L, E\}$. The duplicates 'O' and 'G' are not listed multiple times.

The objects that constitute a set are referred to as its elements or members.

Notation Used for Sets and Elements

• Sets are typically denoted by capital letters of the English alphabet, such as $A, B, C, ..., X, Y, Z$.
• The elements or members of a set are generally denoted by lowercase letters, like $a, b, c, ..., x, y, z$.
• The symbol '$\in$' is used to indicate that an element belongs to a set. If $a$ is an element of set $A$, we write $a \in A$. This is read as "$a$ belongs to $A$", "$a$ is an element of $A$", or "$a$ is a member of $A$".
• The symbol '$\notin$' is used to indicate that an element does not belong to a set. If $b$ is not an element of set $A$, we write $b \notin A$. This is read as "$b$ does not belong to $A$", "$b$ is not an element of $A$", or "$b$ is not a member of $A$".

Fundamental Characteristics of a Set

1. Well-definedness:

   The membership rule must be clear and unambiguous for every potential object.

2. Distinctness:

   Every element in a set is unique. Duplicates are not counted as separate elements.

3. Order of Elements:

   The order in which the elements are listed within the set notation does not matter. For example, the set containing the numbers 1, 5, and 10 is the same whether written as $\{1, 5, 10\}$, $\{5, 10, 1\}$, or $\{10, 1, 5\}$. $\{1, 5, 10\} = \{5, 10, 1\}$.

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Representations of a Set (Roster Form, Set-Builder Form)

After understanding what a set is, the next step is to learn how to represent or describe sets effectively. There are two primary methods used for representing sets: the Roster Form and the Set-Builder Form. Each method has its advantages and is suitable for different types of sets.

1. Roster Form (or Tabular Form)

In the Roster Form, a set is described by listing all its elements explicitly. The elements are written inside curly braces $\{\}$, with each element separated by a comma. This method is straightforward and intuitive, especially for sets with a small number of elements.

Key Characteristics and Conventions of Roster Form:

• All elements of the set are listed.
• The elements are separated by commas.
• The entire list is enclosed within curly braces $\{\}$.
• The order of the elements listed does not affect the set. For example, $\{a, b, c\} = \{b, c, a\}$.
• Elements should not be repeated. If an element appears multiple times in a descriptive collection, it is listed only once in the Roster Form. For example, the set of distinct digits in the number 12231 is $\{1, 2, 3\}$.
• For sets with a large number of elements following a clear pattern, or for infinite sets with a clear pattern, we can use ellipses ($...$) to indicate the continuation of the pattern. For example, the set of the first 50 natural numbers can be written as $\{1, 2, 3, ..., 50\}$. The set of all odd natural numbers can be written as $\{1, 3, 5, ...\}$. However, the pattern must be absolutely clear from the initial elements listed before using ellipses.

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2. Set-Builder Form (or Rule Method)

In the Set-Builder Form, a set is described by stating a common property that all its elements share. This property is not possessed by any object outside the set. This method is particularly useful for describing sets with a large or infinite number of elements where listing every element is impossible or impractical. It concisely defines the set based on the characteristics of its members.

General Representation in Set-Builder Form:

A set $A$ in Set-Builder Form is typically written as:

$A = \{x : P(x)\}$ or $A = \{x \mid P(x)\}$

Understanding the Notation:

• $x$: This is a variable that represents a generic element of the set.
• $:$ (colon) or $\mid$ (vertical bar): Both symbols mean "such that". They separate the variable from the condition.
• $P(x)$: This is the property or rule that $x$ must satisfy to be an element of the set $A$. This property must be well-defined.

The entire expression $\{x : P(x)\}$ is read as "the set of all $x$ such that $x$ has the property $P(x)$".

Often, the Set-Builder Form includes the domain from which the elements are taken to make the definition more precise. For example, $\{x \in \mathbb{Z} \mid x^2 = 9\}$ means "the set of all elements $x$ such that $x$ is an integer ($\mathbb{Z}$) AND the square of $x$ is 9 ($x^2=9$)".

Common sets of numbers used as domains include:

• $\mathbb{N}$: The set of natural numbers $\{1, 2, 3, ...\}$
• $\mathbb{Z}$: The set of integers $\{..., -2, -1, 0, 1, 2, ...\}$
• $\mathbb{Q}$: The set of rational numbers
• $\mathbb{R}$: The set of real numbers
• $\mathbb{C}$: The set of complex numbers

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Comparing Roster Form and Set-Builder Form

Standard Sets of Numbers and Their Notations

In mathematics, certain sets of numbers are used so frequently that they have been given standard names and symbols. Understanding these standard sets and their notations is fundamental to working with sets and other mathematical concepts.

Key Standard Sets of Numbers:

• Natural Numbers ($\mathbb{N}$):

  This is the set of counting numbers. There are varying conventions regarding whether to include zero. In the context often followed in India and many other places, the set of natural numbers starts from 1.

  $\mathbb{N} = \{1, 2, 3, 4, 5, ...\}$

  In some contexts, $\mathbb{N}$ might include 0, denoted as $\mathbb{N}_0$ or $\mathbb{W}$ (Whole Numbers). However, unless otherwise specified, $\mathbb{N}$ usually refers to positive integers starting from 1.

• Whole Numbers ($\mathbb{W}$):

  This set includes all natural numbers along with zero.

  $\mathbb{W} = \{0, 1, 2, 3, 4, ...\}$

  Using set operations, we can express $\mathbb{W}$ in relation to $\mathbb{N}$ (where $\mathbb{N} = \{1, 2, 3, ...\}$):

  $\mathbb{W} = \mathbb{N} \cup \{0\}$

• Integers ($\mathbb{Z}$):

  The set of integers includes all whole numbers and the negative counterparts of the natural numbers. The symbol $\mathbb{Z}$ comes from the German word "Zahlen", meaning numbers.

  $\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

  Integers can be positive ($1, 2, 3, ...$), negative ($-1, -2, -3, ...$), or zero ($0$).

• Rational Numbers ($\mathbb{Q}$):

  This set comprises all numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and the denominator $q$ is not zero. The symbol $\mathbb{Q}$ stands for "quotient".

  In Set-Builder Form:

  $\mathbb{Q} = \left\{ x \mid x = \frac{p}{q}, \text{ where } p \in \mathbb{Z}, q \in \mathbb{Z}, \text{ and } q \neq 0 \right\}$

  Decimal Representation of Rational Numbers:

  When a rational number is expressed in decimal form, the decimal expansion is either terminating (e.g., $\frac{1}{4} = 0.25$) or non-terminating and repeating (e.g., $\frac{1}{3} = 0.333... = 0.\overline{3}$, $\frac{1}{7} = 0.142857142857... = 0.\overline{142857}$). Any number with a terminating or repeating decimal expansion can be written as a fraction $\frac{p}{q}$, thus it is a rational number.

  Relationship with other sets:

  All integers are rational numbers because any integer $n$ can be written as $\frac{n}{1}$, where $n \in \mathbb{Z}$ and $1 \in \mathbb{Z}, 1 \neq 0$. Thus, $\mathbb{Z} \subset \mathbb{Q}$. Since $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z}$, it follows that $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q}$.

• Irrational Numbers ($\mathbb{T}$ or $\mathbb{I}$ or $\mathbb{R} \setminus \mathbb{Q}$):

  This set consists of real numbers that are NOT rational. These numbers cannot be expressed as a fraction $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$.

  Decimal Representation of Irrational Numbers:

  In decimal form, irrational numbers have decimal expansions that are non-terminating and non-repeating. This non-repeating nature is what distinguishes them from rational numbers.

  Examples:

  Famous examples of irrational numbers include $\sqrt{2}, \sqrt{3}, \sqrt{5}$, $\pi$ (pi, the ratio of a circle's circumference to its diameter), and $e$ (Euler's number, the base of the natural logarithm).

• Real Numbers ($\mathbb{R}$):

  The set of real numbers is the union of the set of rational numbers and the set of irrational numbers. The symbol $\mathbb{R}$ stands for "Real". Real numbers represent all points on the number line.

  $\mathbb{R} = \mathbb{Q} \cup \mathbb{T}$ (where $\mathbb{T}$ is the set of irrational numbers)

  Hierarchy of Standard Sets:

  Based on the definitions, we can see a clear hierarchy among these sets using the subset relation ($\subset$):

  $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

  The set of irrational numbers $\mathbb{T}$ is a subset of $\mathbb{R}$ but is disjoint from $\mathbb{Q}$ (i.e., $\mathbb{Q} \cap \mathbb{T} = \emptyset$, where $\emptyset$ is the empty set).

• Complex Numbers ($\mathbb{C}$):

  While not always included in basic set theory discussions related to real numbers, the set of complex numbers is also a standard set. It consists of numbers of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying $i^2 = -1$. The set of real numbers $\mathbb{R}$ is a subset of $\mathbb{C}$ (when $b=0$).

Notations for Subsets of Standard Sets:

Superscripts are often used to denote specific subsets of these standard number sets:

• $\mathbb{Z}^+$: The set of Positive Integers. This is the same as the set of natural numbers starting from 1. $\mathbb{Z}^+ = \{1, 2, 3, ...\} = \mathbb{N}$.
• $\mathbb{Z}^-$: The set of Negative Integers. $\mathbb{Z}^- = \{..., -3, -2, -1\}$.
• $\mathbb{Z}^0$: The set of Non-zero Integers. $\mathbb{Z}^0 = \mathbb{Z} \setminus \{0\}$.
• $\mathbb{W}_0$ or $\mathbb{Z}_{\geq 0}$: The set of Non-negative Integers. This is the same as the set of whole numbers. $\mathbb{W}_0 = \{0, 1, 2, 3, ...\} = \mathbb{W}$.
• $\mathbb{Q}^+$: The set of Positive Rational Numbers. $\mathbb{Q}^+ = \{x \in \mathbb{Q} \mid x > 0\}$.
• $\mathbb{Q}^-$: The set of Negative Rational Numbers. $\mathbb{Q}^- = \{x \in \mathbb{Q} \mid x < 0\}$.
• $\mathbb{R}^+$: The set of Positive Real Numbers. $\mathbb{R}^+ = \{x \in \mathbb{R} \mid x > 0\}$.
• $\mathbb{R}^-$: The set of Negative Real Numbers. $\mathbb{R}^- = \{x \in \mathbb{R} \mid x < 0\}$.

Similarly, $\mathbb{Q}_{\geq 0}$ denotes the set of non-negative rational numbers, and $\mathbb{R}_{\geq 0}$ denotes the set of non-negative real numbers.

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Ordered Pair

In set theory, the order in which elements are listed does not matter. For instance, $\{a, b\}$ and $\{b, a\}$ represent the exact same set. However, in many areas of mathematics, the order of elements is crucial. This is where the concept of an ordered pair becomes essential. An ordered pair is used to group two objects in a specific sequence.

Definition of an Ordered Pair

An ordered pair is a collection of two objects in a specific order. It is denoted by $(a, b)$, where $a$ is called the first component (or first element) and $b$ is called the second component (or second element).

The key difference from a set $\{a, b\}$ is the significance of the position of the elements.

Key Properties of Ordered Pairs

• Order Matters:

  For an ordered pair $(a, b)$, the pair $(b, a)$ is generally considered different unless $a = b$. This is unlike sets where $\{a, b\} = \{b, a\}$ always holds.

  Illustration:

  Consider points on a coordinate plane. The ordered pair $(2, 3)$ represents a point located 2 units along the x-axis and 3 units along the y-axis. The ordered pair $(3, 2)$ represents a different point, located 3 units along the x-axis and 2 units along the y-axis.

  

  (Imagine an image showing the points (2,3) and (3,2) plotted on a standard Cartesian coordinate system to visually demonstrate they are different positions).

• Equality of Ordered Pairs:

  Two ordered pairs $(a, b)$ and $(c, d)$ are defined to be equal if and only if their corresponding first components are equal AND their corresponding second components are equal.

  $(a, b) = (c, d)$

  if and only if

  $a = c \text{ and } b = d$

  This property is fundamental for solving equations involving ordered pairs, as shown in the example below.

• Components can be Identical:

  The components of an ordered pair do not have to be distinct. The pair $(a, a)$ is a valid ordered pair. For example, $(5, 5)$ or $(m, m)$ are perfectly valid ordered pairs.

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Cartesian Product of Two Sets

Building upon the concept of ordered pairs, we can define an operation between two sets that produces a new set consisting of all possible ordered pairs where the first element comes from the first set and the second element comes from the second set. This operation is called the Cartesian Product.

Definition of Cartesian Product

Let $A$ and $B$ be two non-empty sets. The Cartesian product of set $A$ and set $B$, denoted by $A \times B$ (read as "A cross B"), is the set of all possible ordered pairs $(a, b)$ such that the first component $a$ is an element of set $A$, and the second component $b$ is an element of set $B$.

In Set-Builder Form, the Cartesian product is defined as:

If either set $A$ or set $B$ is the empty set ($\emptyset$), their Cartesian product is the empty set. This is because it's impossible to form an ordered pair $(a, b)$ if there are no elements in $A$ or no elements in $B$.

Key Properties and Characteristics of Cartesian Product

• The Result is a Set of Ordered Pairs:

  The elements of the Cartesian product $A \times B$ are not individual elements from $A$ or $B$, but rather the ordered pairs formed by taking one element from $A$ and one from $B$ in that specific order.

• Order of Sets Matters:

  In general, the Cartesian product $A \times B$ is not equal to the Cartesian product $B \times A$.

  $A \times B \neq B \times A$ (generally)

  This is because the ordered pairs in $A \times B$ have the first element from $A$ and the second from $B$, while the ordered pairs in $B \times A$ have the first element from $B$ and the second from $A$. An ordered pair $(a, b)$ is equal to $(b, a)$ only if $a=b$. The sets $A \times B$ and $B \times A$ are equal if and only if $A = B$ or if at least one of the sets is empty.

  $A \times B = B \times A \iff A = B \text{ or } A = \emptyset \text{ or } B = \emptyset$

• Cardinality of the Cartesian Product:

  The number of elements in the Cartesian product of two finite sets is the product of the number of elements in each set.

  If $|A|$ (the number of elements in set A) is $p$, and $|B|$ (the number of elements in set B) is $q$, then the number of elements in $A \times B$ is $p \times q$.

  $|A \times B| = |A| \times |B| = pq$

  This also implies that $|B \times A| = |B| \times |A| = qp$, which is equal to $pq$. So, $A \times B$ and $B \times A$ always have the same number of elements, even if they are not the same set.

• Geometric Interpretation:

  The Cartesian product is named after René Descartes, whose work led to the development of coordinate geometry. The Cartesian plane (the xy-plane) is essentially the Cartesian product of the set of real numbers with itself, $\mathbb{R} \times \mathbb{R} = \{ (x, y) \mid x \in \mathbb{R}, y \in \mathbb{R} \}$. Each point on the plane corresponds to a unique ordered pair of real numbers.

• Cartesian Product of Three or More Sets:

  The concept extends to the Cartesian product of three or more sets. The Cartesian product $A \times B \times C$ of three sets $A, B, C$ is the set of all ordered triplets $(a, b, c)$ where $a \in A$, $b \in B$, and $c \in C$.

  $A \times B \times C = \{ (a, b, c) \mid a \in A, b \in B, c \in C \}$

  If $|A|=p, |B|=q, |C|=r$, then the number of elements in $A \times B \times C$ is $|A \times B \times C| = |A| \times |B| \times |C| = pqr$. Similarly, $A \times A = A^2 = \{ (a_1, a_2) \mid a_1 \in A, a_2 \in A \}$ and $A \times A \times A = A^3 = \{ (a_1, a_2, a_3) \mid a_1 \in A, a_2 \in A, a_3 \in A \}$, etc.

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