Sets

Introduction

Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets.

The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”.

Set, in mathematics and logic, any collection of objects. The objects in a set are called its elements or members.

Set : A well-defined collection of objects is called a set.

We denote sets by capital letters A, B, C, X, Y, Z, etc, and denote their generic elements by their corresponding lowercase letters a, b, c, s and t, respectively.

If a is an element of a set A, we write, ad A, which means that a belongs to A or that a is an element of A. If a does not belong to A, we write, A.

Sets and their Representations

In everyday life, we often speak of collections of objects of a particular kind, such as, a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come across collections, for example, of natural numbers, points, prime numbers, etc.

More specially, we examine the following collections: the sense that we can definitely decide whether a given particular object belongs to a given collection or not. For example, we can say that the river Nile does not belong to the collection of rivers of India. On the other hand, the river Ganga does belong to this colleciton.

We give below a few more examples of sets used particularly in mathematics, viz.

• ℕ : the set of all natural numbers
• ℤ or 𝕀 : the set of all integers
• ℚ : the set of all rational numbers
• ℝ : the set of real numbers
• ℤ⁺ : the set of positive integers
• ℚ⁺ : the set of positive rational numbers, and
• ℝ⁺ : the set of positive real numbers.

The symbols for the special sets given above will be referred to throughout this text. Again the collection of five most renowned mathematicians of the world is not well-defined, because the criterion for determining a mathematician as most renowned may vary from person to person. Thus, it is not a well-defined collection.

There are two methods of representing a set :

1. Roster or tabular form
2. Set-builder form.

Types of Sets

1. Empty set or Void set : A set which does not contain any element is called empty set.
2. Finite set and Infinite set : A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set.
3. Equal sets : Two sets A and B are said to be equal if they have exactly the same elements.
4. Subsets : A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R.
5. Power sets: A power set of a set A is collection of all subsets of A. It is denoted by P(A)
6. Universal set

1. Empty Sets

A set which does not contain any element is called the empty set or the void set or null set and is denoted by { } or φ.

A set which has at least one element is called a non-empty set.

Examples of empty sets :

1. {x : x ∈ N and 2 < x < 3} = φ, since there is no natural number lying between 2 and 3.
2. {x : x ∈ R and x² = -1} = φ, since there is no real number whose square is –1.
3. {x : x is an even prime number greater than 2} = φ, since there is no prime number which is even and greater than 2.
4. D = { x : x² = 4, x is odd }. Then D is the empty set, because the equation x² = 4 is not satisfied by any odd value of x.

2. Finite and Infinite Sets

An empty set or a nonempty set in which the process of counting of elements surely comes to an end is called a finite set. A set which is not finite is called an infinite set.

Examples of finite sets :

1. The set of all persons on earth is a finite set.
2. The set of all animals on earth is a finite set.
3. Let A = {2, 4, 6, 8, 10, 12}.
   Then, A is clearly a finite set and n(A) = 6.
4. Let B = set of all letters in the English alphabet.
   Then, n(B) = 26 and therefore, B is finite.

Examples of infinite sets :

1. The set of all straight lines parallel to a given line, say the x-axis, is an infinite set.
2. The set of all positive integral multiples of 5 is an infinite set.
   Let Z⁺ be the set of all positive integers.
   Then, {5x : x ∈ Z⁺} = {5, 10, 15, 20, 25, …} is an infinite set.

3. Equal Sets

Two nonempty sets A and B are said to be equal, if they have exactly the
same elements and we write, A = B.

1. The elements of a set may be listed in any order.
   Thus, {1, 2, 3} = {2, 1, 3} = {3, 2, 1}
2. The repetition of elements in a set has no meaning.
   Thus, {1, 1, 2, 2, 3} = {1, 2, 3}

Examples of equal sets :

Equivalent Sets

Two finite sets A and B are said to be equivalent, if n(A) = n(B) . Equal sets are always equivalent. But, equivalent sets need not be equal.

Examples of equivalent sets :

4. Subsets

A set A is said to be a subset of set B if every element of A is also an element
of B, and we write, A ⊆ B.

If there exists even a single element in A which is not in B, then A is not a subset of B, and we write, A 1

Examples of subset :

Superset

If A ⊆ B, then B is called a superset of A, and we write, B ⊇ A.

Proper Subset

If A ⊆ B and A ≠ B then A is called a proper subset of B and we, A ⊂ B.

If there exists even a single element in A which is not in B, then A is not a subset of B, and we write, A ⊂ B.

5. Power sets

The set of all subsets of a given set A is called the power set of A, denoted by P(A) .

If n(A) = m then n[P(A)] = 2^m.

6. Universal Set

If some sets are under consideration, then there happens to be a set that is a superset of each of the given sets. Such a set is known as the universal set for those sets. We shall denote a universal set by U.

Usually, in a particular context, we have to deal with the elements and subsets of a basic set that is relevant to that specific context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “Universal Set“. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers. For another example, in human population studies, the universal set consists of all the people in the world.

Operations on Sets

In earlier classes, we have learnt how to perform the operations of addition, subtraction, multiplication and division on numbers. Each one of these operations was performed on a pair of numbers to get another number. For example, when we operate addition on the pair of numbers 5 and 13, we get the number 18. Again, operating multiplication on the pair of numbers 5 and 13, we get 65. Similarly, there are some operations that when performed on two sets give rise to another set. We will now define certain operations on sets and examine their properties. Henceforth, we will refer to all our sets as subsets of some universal set.

Type of Operations on sets:

1. Union of sets: The union of two sets A and B is the set of all those elements which are either in A or in B.
2. Intersection of sets: The intersection of two sets A and B is the set of all common elements.
3. Difference of sets: The difference between two sets A and B in this order is the set of elements that belong to A but not to B.
4. Complement of a set: The complement of a subset A of universal set U is the set of all elements of U that are not the elements of A.

Union of Sets

The union of two sets A and B, denoted by A , B, is the set of all those elements which are either in A or in B or in both A and B. Thus,

A ∪ B = {x : x ∈ A or x ∈ B}

x ∈ A ∪ B ⇔ x ∈ A or x ∈ B.

Example:

Intersection of Sets

The intersection of two sets A and B, denoted by A + B, is the set of all elements which are common to both A and B.

Thus,

A ∩ B = {x : x ∈ A and x ∈ B}

∴ x ∈ A ∩ B ⇔ x ∈ A and x ∈ B

x ∉ A ∩ B ⇒ x ∉ A or x ∉ B.

Example:

Disjoint Sets

Two sets A and B are said to be disjoint if A ∩ B = φ

Difference of Sets

For any sets A and B, their difference (A − B) is defined as

(A − B) = {x : x ∈ A and x ∉ B}

Thus, x ∈ (A − B) => x ∈ A and x ∉ B.

Symmetric Difference of Two Sets

The symmetric difference of two sets A and B, denoted by A ∆ B, is defined as

A ∆ B = (A − B) ∪ (B − A)

Complement of a Set

Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U − A), is defined as

A’ = { x ∈ U : x ∉ A }

Clearly, x A’ <=> x A.

Various Laws of Operations on Sets

1. Idempotent laws :
   1. A ∪ A = A
   2. A ∩ A = A
2. Identity laws :
   1. A ∪ φ = A
   2. A ∩ U = A
3. Commutative laws :
   1. A ∪ B = B ∪ A
   2. A ∩ B = B ∩ A
4. Associative laws :
   1. (A ∪ B) ∪ C = A ∪ (B ∪ C)
   2. (A ∩ B) ∩ C = A ∩ (B ∩ C)
5. Distributive law :
   1. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
   2. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
6. De Morgan‘s laws :
   1. (A ∪ B)’ = (A’ ∩ B’)
   2. (A ∩ B)’ = (A’ ∪ B’)

Venn Diagrams

Most of the relationships between sets can be represented using diagrams which are known as Venn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles.

To express the relationship among sets in perspective, we represent them pictorially using diagrams, called Venn diagrams. In these diagrams, the universal set is represented by a rectangular region and its subsets by circles inside the rectangle. We represent disjoint sets by disjoint circles and intersecting sets by intersecting circles.

Venn Diagrams in Different Situations

To express the relationship among sets in perspective, we represent them pictorially using diagrams, called Venn diagrams. In these diagrams, the universal set is represented by a rectangular region and its subsets by circles inside the rectangle. We represent disjoint sets by disjoint circles and intersecting sets by intersecting circles.

Important Results from Venn Diagrams

In the earlier Section, we have learnt union, intersection and difference of two sets. In this Section, we will go through some practical problems related to our daily life.

Let A and B be two intersecting subsets of U. In counting the elements of (A ∪ B), the elements of A + B are counted twice, once in counting the elements of A and a second time in counting the elements of B.

∴ n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

If A ∩ B = , then n(A ∩ B) = 0 and therefore, in this case, we have

∴ n(A ∪ B) = n(A) + n(B)

From the Venn diagram, it is also clear that

1. n(A − B) + n(A ∩ B) = n(A)
2. n(B − A) + n(A ∩ B) = n(B)
3. n(A − B) + n(A ∩ B) + n(B − A) = n(A ∪ B).

Historical Reading

The modern theory of sets is considered to have originated largely by the German mathematician Georg Cantor (1845-1918). His papers on set theory appeared sometimes from 1874 to 1897. His study of set theory came when he was studying trigonometric series of the form

a₁ sin x + a₂ sin 2x + a₃ sin 3x + …

He published in a paper in 1874 that the set of real numbers could not be put into one-to-one correspondence with the integers. From 1879 onwards, he published several papers showing various properties of abstract sets.

Cantor’s work was well received by another famous mathematician Richard Dedekind (1831-1916). But Kronecker (1810-1893) castigated him for regarding infinite set the same way as finite sets. Another German mathematician Gottlob Frege, at the turn of the century, presented the set theory as principles of logic. Till then the entire set theory was based on the assumption of the existence of the set of all sets. It was the famous English Philosopher Bertrand Russell (1872-1970 ) who showed in 1902 that the assumption of the existence of a set of all sets leads to a contradiction. This led to the famous Russell’s Paradox. Paul R. Holmos writes about it in his book ‘Naïve Set Theory that “nothing contains everything”.

Russell’s Paradox was not the only one that arose in set theory. Many paradoxes were produced later by several mathematicians and logicians. As a consequence of all these paradoxes, the first axiomatisation of set theory was published in 1908 by Ernst Zermelo. Another one was proposed by Abraham Fraenkel in 1922. John Von Neumann in 1925 introduced explicitly the axiom of regularity. Later in 1937, Paul Bernays gave a set of more satisfactory axiomatisation. A modification of these axioms was done by Kurt Gödel in his monograph in 1940. This was known as Von Neumann-Bernays (VNB) or Gödel-Bernays (GB) set theory.

Despite all these difficulties, Cantor’s set theory is used in present-day mathematics. These days most of the concepts and results in mathematics are expressed in the set theoretic language.