Definitions

Definition of Sets

A set is a fundamental concept in mathematics, defined as a collection of distinct objects, treated as an entity in itself. These objects are known as the elements or members of the set.

Basic Representation of sets

The basic representation of sets in mathematics uses curly braces to enclose a collection of distinct, well-defined objects. These objects are the elements of the set. The use of curly braces is a universally accepted convention in mathematics to denote a set.

Within the curly braces, the elements of the set are listed, typically separated by commas. For example, a set of the first three natural numbers is represented as \(\{1, 2, 3\}\). The elements inside the set are distinct; duplicates are not acknowledged in a set. So, if a collection includes duplicates, like 2, 2, 3, it's represented as \(\{2, 3\}\) in set notation, eliminating any repetitions.

Another important characteristic of sets is that the order of elements does not matter. Therefore, \(\{1, 2, 3\}\) is the same as \(\{3, 2, 1\}\). This unordered nature is a defining feature of set theory.

For clarity, here are five examples of sets represented using this notation:

A set of primary colors: \(\{red, blue, yellow\}\).
A set of single-digit even numbers: \(\{2, 4, 6, 8\}\).
A set of vowels in the English alphabet: \(\{a, e, i, o, u\}\).
A set of basic geometric shapes: \(\{circle, square, triangle\}\).
A set containing a mix of numbers and letters: \(\{1, x, 3, y\}\).

In each of these examples, the set is defined by the unique elements it contains, irrespective of their order, and is enclosed within curly braces, following the standard set notation in mathematics.

Characteristics of a Set

Unordered: The elements in a set do not have a specific order. For example, the set containing numbers 1, 2, and 3 is written as \(1, 2, 3\) and is the same as \(3, 2, 1\).
Well-Defined: A set is well-defined when it is clear whether any particular item belongs to the set or not. For instance, the set of all letters in the word 'apple' is well-defined as \(a, p, l, e\).
Distinctness: Each element in a set is unique. If an object is in a set, it cannot be in the set more than once. For example, in the set \(a, b, c\), each letter is distinct and appears only once.

Examples of Sets

A set of prime numbers less than 10: \(\{2, 3, 5, 7\}\).
A set of colors in a rainbow: \(\{red, orange, yellow, green, blue, indigo, violet\}\).
A set of months in a year: \(\{January, February, March, ..., December\}\).
Fibonacci Numbers Less Than 30: \(\{0, 1, 1, 2, 3, 5, 8, 13, 21\}\)
This set consists of Fibonacci numbers up to 30.
First Five Positive Even Numbers: \(\{2, 4, 6, 8, 10\}\)
A set of the smallest even numbers.
Primary Colors in the RGB Color Model: \(\{Red, Green, Blue\}\)
The set of primary colors used in digital color representation.
Basic Tastes Detected by Human Tongue: \(\{Sweet, Sour, Salty, Bitter, Umami\}\)
This set includes all the primary tastes our taste buds can detect.
Planets in Our Solar System: \(\{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune\}\)
A set containing the names of all planets that orbit the Sun.
First Five Roman Numerals: \(\{I, II, III, IV, V\}\)
This set represents the first five numbers in the Roman numeral system.
Continents on Earth: \(\{Africa, Antarctica, Asia, Europe, North America, Oceania, South America\}\)
The set of all seven continents in the world.
Chess Pieces for One Player: \(\{King, Queen, Bishop, Knight, Rook, Pawn\}\)
A set representing each type of chess piece in a standard chess game.
Olympic Rings Colors: \(\{Blue, Yellow, Black, Green, Red\}\)
The set of colors used in the Olympic rings symbol.
Basic Geometric Shapes: \(\{Circle, Square, Triangle, Rectangle, Pentagon, Hexagon\}\)
This set contains different simple geometric shapes.

Not all collections are sets

Not all collections are sets.Collections that are not considered sets typically violate fundamental properties of set theory, such as the uniqueness and well-defined nature of elements. Here are examples of such collections:

A Collection with Duplicate Elements: Example: \(\{\text{apple}, \text{banana}, \text{apple}\}\)
This collection includes 'apple' twice, violating the principle of unique elements in a set.
A Collection with Ambiguous Elements: Example: \(\{\text{the tallest building}, \text{the fastest car}\}\)
The elements are not well-defined as they can change over time or depend on context.
A Collection with Vague Criteria: Example: \(\{\text{interesting numbers}, \text{beautiful shapes}\}\)
'Interesting' and 'beautiful' are subjective terms, making this collection not well-defined.
A Collection Based on Uncertain Membership: Example: \(\{\text{future Nobel laureates}, \text{next decade's bestselling authors}\}\)
The members of this collection are unknown and speculative, not fitting the well-defined criterion.
A Collection with Non-Distinct Categories: Example: \(\{\text{fruit}, \text{apple}, \text{orange}\}\)
'Fruit' is a category that includes 'apple' and 'orange', making the collection non-distinct.
A Continuously Changing Collection: Example: \(\{\text{current stock market leaders}\}\)
This is not a set because its composition changes frequently.
A Collection with Incomparable Elements: Example: \(\{2, \text{"two"}, \{2\}\}\)
This collection mixes numbers, strings, and sets, making it inconsistent in terms of element type comparison.

Each of these examples demonstrates collections that fail to meet the criteria for sets in set theory, either due to the presence of duplicates, lack of well-defined elements, or inconsistency in the nature of the elements.

Multiset

A multiset, also known as a bag, is a collection of elements where duplicates are allowed. Unlike a traditional set, in a multiset, the same element can appear more than once. The order of elements in a multiset is not significant. Multisets are useful in situations where the frequency of elements is important.

Set Membership Symbol

The set membership symbol in set theory is \(\in\). It is used to indicate that a particular element is a part of a set. Conversely, the symbol \(\notin\) denotes that an element is not a part of a set.

Examples:

For a Set of Natural Numbers:
Let's say we have a set \(N = \{1, 2, 3, 4, 5\}\).
We can state that \(3 \in N\), which means 3 is a member of the set N.
Similarly, \(6 \notin N\) signifies that 6 is not a member of the set N.
For a Set of Vowels:
Consider a set \(V = \{a, e, i, o, u\}\).
Here, \(e \in V\) indicates that e is a member of the set V.
On the other hand, \(b \notin V\) indicates that b is not a member of the set V.

These examples demonstrate how the symbols \(\in\) and \(\notin\) are used in set theory to express membership and non-membership, respectively.

Special Sets

Some sets are special and we refer to them again and again. So we give them special symbols.

Natural Numbers (ℕ): Symbol: \(\mathbb{N}\)
Description: The set of natural numbers includes all positive integers starting from 1. In some definitions, it also includes 0.
Example: \(\{1, 2, 3, 4, ...\}\)
Whole Numbers: Symbol: Often denoted as \(\mathbb{W}\) or \(\mathbb{N}_0\), but there is no universal symbol.
Description: Whole numbers include all natural numbers along with 0.
Example: \(\{0, 1, 2, 3, 4, ...\}\)
Integers (ℤ): Symbol: \(\mathbb{Z}\)
Description: The set of integers encompasses all whole numbers as well as their negative counterparts.
Example: \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\)
Rational Numbers (ℚ): Symbol: \(\mathbb{Q}\)
Description: Rational numbers include all fractions and integers, essentially numbers that can be expressed as the quotient of two integers.
Example: Numbers like \(1/2\), \(4/5\), \(7\), \(-3\), etc.
Real Numbers (ℝ): Symbol: \(\mathbb{R}\)
Description: Real numbers consist of all rational and irrational numbers, covering every point on the number line.
Example: Includes numbers like \(1.414\), \(\pi\), \(e\), \(3/4\), \(5\), etc.
Complex Numbers (ℂ): Symbol: \(\mathbb{C}\)
Description: Complex numbers include all numbers that can be written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
Example: Numbers like \(3 + 4i\), \(-2 - i\), etc.
Positive Real Numbers (ℝ⁺): Symbol: \(\mathbb{R}^+\)
Description: Includes all real numbers greater than 0.
Example: Numbers like \(0.5\), \(3.14\), and \(e\).
Positive Integers (ℕ⁺): Symbol: \(\mathbb{N}^+\) or \(\mathbb{Z}^+\)
Description: Contains all positive integers, starting from 1.
Example: \(\{1, 2, 3, 4, ...\}\)
Non-Negative Integers: Symbol: Often \(\mathbb{N}_0\) or \(\mathbb{Z}_{\geq 0}\)
Description: Includes all positive integers along with 0.
Example: \(\{0, 1, 2, 3, ...\}\)
Negative Integers (ℤ⁻): Symbol: \(\mathbb{Z}^-\)
Description: The set of all negative integers.
Example: \(\{..., -4, -3, -2, -1\}\)
Positive Rational Numbers: Symbol: \(\mathbb{Q}^+\)
Description: All rational numbers greater than 0.
Example: Numbers like \(\frac{1}{2}\), \(\frac{3}{4}\), \(5\).
Negative Rational Numbers: Symbol: \(\mathbb{Q}^-\)
Description: The set of all rational numbers less than 0.
Example: \(-\frac{1}{2}\), \(-\frac{4}{5}\), \(-3\).
Irrational Numbers: Symbol: There is no universally accepted symbol, but sometimes denoted as \(\mathbb{I}\).
Description: Comprises real numbers that cannot be expressed as a fraction of two integers.
Example: Numbers like \(\sqrt{2}\), \(\pi\), \(e\).

Empty set

The empty set, also known as the null set, is a basic concept in set theory. It is the unique set that contains no elements. The common symbols used to represent the empty set are \( \emptyset \) and \( \{ \} \).

Origin

The symbol \( \emptyset \) was introduced by the Danish mathematician and logician, André Weil, in the mid-20th century. He was inspired by the Norwegian alphabet, which includes the letter Ø, symbolizing the concept of nothing or emptiness. This symbol has since become standard in mathematics to represent a set with no elements.

For more detailed information about the empty set and its origins, you can refer to its Wikipedia page: Empty Set.

Set Builder form

Till now, when we were expressing sets in roster form, that is, by explicitly listing out all the elements of the set, we were using a straightforward and clear method suitable for smaller or well-defined collections. However, when dealing with larger or more complex sets, this approach can become impractical. In such cases, the set-builder form of expressing sets is more efficient and useful.

Set-Builder Form Explained:

Set-builder form is a method of defining sets that specifies the properties or rules that the members of the set must satisfy, rather than listing each element individually. This form is particularly useful for describing sets with an infinite number of elements or sets defined by a specific mathematical rule or property. The general notation for set-builder form is:

\[ \{ x \mid \text{property}(x) \} \]

In this notation:

\( x \) represents an element of the set.
\(\text{property}(x)\) is a statement or condition that defines the set, describing a property that elements \( x \) must satisfy to be included in the set.
The vertical bar \(\mid\) (or sometimes a colon \(:\)) is read as "such that" or "for which".

For instance,

\[ \{ x \in \mathbb{R} \mid x^2 - 4x + 3 = 0 \} \]

In this set, \( x \) represents an element of the set, and the condition \( x^2 - 4x + 3 = 0 \) is an equation that elements of the set must satisfy. The set includes all real numbers \( x \) that are solutions to this quadratic equation.

More Examples of Set-Builder Form:

Set of All Even Numbers: \(\{ x \in \mathbb{N} \mid x \text{ is even} \}\)
This defines a set consisting of all even natural numbers.
Set of Positive Integers Less than 10: \(\{ x \in \mathbb{N} \mid x < 10 \}\)
Here, the set includes all natural numbers less than 10.
Set of Real Numbers Greater than Zero: \(\{ x \in \mathbb{R} \mid x > 0 \}\)
This set consists of all real numbers greater than zero.
Set of Ordered Pairs Where the Sum Equals 10: \( \{ (x, y) \in \mathbb{R}^2 \mid x + y = 10 \} \)
This set includes all ordered pairs \((x, y)\) of real numbers where the sum of \(x\) and \(y\) is equal to 10.
Set of Ordered Pairs Where the Product Equals 0: \( \{ (x, y) \in \mathbb{R}^2 \mid xy = 0 \} \)
This set consists of all ordered pairs \((x, y)\) of real numbers where the product of \(x\) and \(y\) equals 0.

Conciseness of Set-builder form

Observe the following example: \( \left\{ \frac{n}{n+1} \mid n \in \mathbb{N} \right\} \). In this set, each element is formed by the expression \( \frac{n}{n+1} \), where \( n \) is a natural number. The set includes all such fractions starting from \( n = 1 \) onwards, representing an infinite sequence of fractions that approach 1. Set-builder notation is a concise and flexible way to define sets, especially when the set is infinite or follows a clear pattern defined by the given property or rule.

Subset and Superset

Subset

A subset of a set is defined as follows:

Given two sets \(A\) and \(B\), set \(A\) is considered a subset of set \(B\) if and only if every element \(x\) in set \(A\) is also an element of set \(B\). This relationship is denoted as:

\[ A \subseteq B \]

In this notation:

\(A\) is the subset set.
\(B\) is the parent set.
The symbol \(\subseteq\) represents the subset relationship.

In simple terms, set \(A\) is a subset of set \(B\) if all the elements of set \(A\) are contained within set \(B\). It's important to note that a set can also be considered a subset of itself, as every element of \(A\) is also in \(A\).

For example, if we have sets: \(A = \{1, 2, 3\}\) and \(B = \{1, 2, 3, 4, 5\}\), then \(A\) is a subset of \(B\) because every element in \(A\) is also in \(B\), and we can write this as \(A \subseteq B\).

Here are examples of subsets, both common and creative, using Markdown with LaTeX:

Set of Natural Numbers as a Subset of Integers: Let \( \mathbb{N} \) represent the set of natural numbers (\( \mathbb{N} = \{1, 2, 3, \ldots\} \)) and \( \mathbb{Z} \) represent the set of integers (\( \mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\} \)).
Mathematically: \( \mathbb{N} \subseteq \mathbb{Z} \), indicating that every natural number is also an integer.
Set of Prime Numbers as a Subset of Natural Numbers: Let \( P \) represent the set of prime numbers (\( P = \{2, 3, 5, 7, 11, \ldots\} \)) and \( \mathbb{N} \) represent the set of natural numbers.
Mathematically: \( P \subseteq \mathbb{N} \), indicating that every prime number is a natural number.
Set of Mammals as a Subset of Animals: Let \( M \) represent the set of mammals and \( A \) represent the set of all animals.
Mathematically: \( M \subseteq A \), indicating that all mammals are animals.

Proper subset

Given two sets \(A\) and \(B\), set \(A\) is considered a proper subset of set \(B\) if and only if every element \(x\) in set \(A\) is also an element of set \(B\), and \(A\) is not equal to \(B\). This relationship is denoted as:

\[ A \subset B \]

In this notation:

\(A\) is the proper subset.
\(B\) is the parent set.
The symbol \(\subset\) represents the proper subset relationship.

In simpler terms, set \(A\) is a proper subset of set \(B\) if all the elements of set \(A\) are contained within set \(B\), and \(A\) is not the same as \(B\). The symbol \(\subset\) is used to indicate that it is a proper subset, implying that it is a subset but not equal to the parent set.

Superset

We used the term 'parent' above. Instead of this term, we have a better term 'superset' defined as follows:

Given two sets \(A\) and \(B\), set \(B\) is considered a superset of set \(A\) if and only if every element \(x\) in set \(A\) is also an element of set \(B\). This relationship is denoted as:

\[ B \supseteq A \]

In this notation:

\(B\) is the superset.
\(A\) is the subset.
The symbol \(\supseteq\) represents the superset relationship.

Examples:

Superset of Natural Numbers: Let \( \mathbb{Z} \) represent the set of integers (\( \mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\} \)) and \( \mathbb{N} \) represent the set of natural numbers (\( \mathbb{N} = \{1, 2, 3, \ldots\} \)). Mathematically: \( \mathbb{Z} \supseteq \mathbb{N} \).
Superset of Primary Colors: Let \( P = \{\text{Red, Blue, Yellow}\} \) represent the set of primary colors, and \( C \) represent the set of all colors. Mathematically: \( C \supseteq P \).
Superset of Vowels: Let \( V = \{a, e, i, o, u\} \) represent the set of vowels in the English alphabet, and \( E \) represent the set of all English alphabets. Mathematically: \( E \supseteq V \).

Cardinality

The cardinality of finite sets refers to the number of elements or members in a finite set. It is denoted as \(\left| A \right|\) or \(\text{card}(A)\) or \(n(A)\), where \(A\) represents the finite set. Essentially, it provides a count of how many items are present in the set.

Examples:

If we have the set \(S = \{1, 2, 3, 4, 5\}\), then the cardinality of \(S\) is \(\left| S \right| = 5\).
For the set of months in a year, \(M = \{\text{January, February, March, ..., December}\}\), the cardinality of \(M\) is \(\left| M \right| = 12\).
Consider a set \(F = \{x \in \mathbb{N} \mid 1 \leq x \leq 100\}\), which represents the first 100 natural numbers. The cardinality of \(F\) is \(\left| F \right| = 100\).

Cardinality of Infinite Sets

The cardinality of infinite sets is a concept that can be a bit more complex. In simple terms, it's a way to compare the "size" of different infinite sets to see if they have the same number of elements, even though they are infinite.

One way to explain this idea is to use the concept of "countability." Some infinite sets, like the set of natural numbers (\( \mathbb{N} = \{1, 2, 3, \ldots\} \)), are countable because you can list their elements one by one, even though they go on forever. Other sets, like the set of all real numbers (\( \mathbb{R} \)), are uncountable because you can't list all of their elements in a sequence.

For a more detailed explanation and examples, you can refer to the Wikipedia page on Cardinality.

Types of Sets:

a. Empty Set (Null Set or Void Set): This set contains no elements and is denoted by \( \emptyset \) or \( \phi \). Note that \( \{\emptyset\} \) is not an empty set but a set containing one element, which itself is an empty set.

b. Singleton Set: A set with exactly one element. For instance, \( A = \{\emptyset\} \), \( B = \{e\} \).

d. Finite Set: A set with a limited number of elements. For example, \( \{a, e, i, o, u\} \) represents a set of vowels with a finite number of elements. Similarly, \( \{\mathbb{Z}, \mathbb{N}, \mathbb{R}, \mathbb{Q}\} \) constitutes a set with four elements.

e. Infinite Set: A set with an unbounded number of elements. An example is the set of all real numbers \( \mathbb{R} \), or the set of all natural numbers. The set \( \{x \in \mathbb{R} | 1 < x < 2\} \) is an example of a set of real numbers between 1 and 2.

f. Universal Set: In any context where sets are being considered, there exists a set that encompasses all elements under discussion, known as the universal set. It includes every element from each set in the given context and may also contain additional elements not present in any of the sets under consideration. For example, given the sets \( A = \{1,2\} \), \( B = \{3,4,5,6\} \), and \( C = \{6,7\} \), the universal set could be \( U = \{0,1,2,3,4,5,6,7,8,9\} \), containing all elements from sets \( A \), \( B \), and \( C \), as well as elements like 0 and 9, which are not in \( A \), \( B \), or \( C \).

As an additional example, the set of real numbers \( \mathbb{R} \) can serve as a universal set for the collection including the integers \( \mathbb{Z} \), natural numbers \( \mathbb{N} \), and rational numbers \( \mathbb{Q} \).