Basics of Set Theory

Developed by German mathematician and logician Georg Cantor, Set Theory has massive applications in various branches of mathematics, such as functions, relations, probability etc.

In this article, we will try to cover all the basics of Set Theory.

What is a Set?

A set is a well-defined collection of numbers, perople or objects, which are called its members or elements.

We generally use a capital letter to denote a set, e.g. A, B, S etc. While to denote an element of a set, we use small letters, e.g. a, b, s etc.

If an element, say a, belongs to a certain set, say A, then we denote this as: a ∈ A
If an element, say a, does not belong to a certain set, say A, then we denote this as: a ∉ A

Ways of representation of a Set

There a couple of ways via which we can represent a set.

Roster or Tabular form

In this form of representation, we just write down all the elements of a set (separated by a comma) within curly brackets { }.

For example:

• {2, 4, 6, ….} - set of even numbers. This set has infinite number of elements.
• {a, e, i, o, u} - set of vowels in English alphabet. This set has finite number of elements, i.e. 5.

Set-builder form

We cannot use Roster form for very large sets. Hence, mathematicians came up with another method of representing a set.

In set-builder form, we represnt a single common property which is possessed by:

• all the elements of that set, and
• only by the elements of that set (and no other element outside the set).

A(x) = {x: x has xyz property}
We read this as: A is a set of all x, such that they all fulfil xyz property.

For example:

• A(x) = {x: x is an even number} - It means A is a set of all even numbers.
• A(x) = {x: x is a vowel in English alphabet} - It means A is a set of a, e, i, o, u.

Types of Sets

There are various types of sets that you must be aware of.

Finite and Infinite Sets

A finite set contains a definite number of elements. For example, {2, 4, 5}

On the other hand, an infinite set contains indefinite number of elements. For example, set of points within a circle.

Empty Set

An empty set (or null set, or void set) is a set that does not have any element. It is denoted by the symbol ∅, or just by empty curly brackets { }.

For example, A = {x: x is a prime number greater than 13 but smaller than 17}. As there is no prime number between 13 and 17, so A is an empty set.

Subsets and Power Set

A set is a subset of another set, if all of its elements are also present in the other set. That is, if if a ∈ A ⇒ a ∈ B, then it means A ⊂ B.

For example, if A = {1, 2, 3} and B = {1, 2, 3, 4}, then A is the subset of B. We can see here, that all the elements of set A are also present in set B.

Power set of a set is the collection of all of its subsets (including the empty set). It is denoted by P.

For example, if we have a set A = {H, T}, then its power set P(A) = {∅, {H}, {T}, {H, T}}

Universal Set

A universal set contains all the elements in a given context (that may be part of other sets or not). It is denoted by U.

For example, when we roll a dice, we can get any number from 1 to 6. So, the set {1, 2, 3, 4, 5, 6} is the universal set in this context.

All other sets will be its subsets. For example, A = {x: x is an odd number on the dice}, i.e. {1, 3, 5} is a subset of the universal set {1, 2, 3, 4, 5, 6}.

Relationship between Sets

We use Venn diagrams to show relationship between two or more sets.

Disjoint Sets

Two sets can be called disjoint sets, if they have no common element. We can represent them using Venn diagram as follows:

Overlapping Sets

If there are some common elements in set A and B, then their Venn diagrams will overlap and we can represent their relation as follows:

Equal Sets

Two sets are said to be equal to each other, if they have exactly the same elements. That is, when two sets are identical.

They are denoted by = sign. For example, if set A is identical to set B, then we can write A = B.

In terms of Venn diagram, we can say that their Venn diagrams will completely overlap.

Operations on Sets

Now, we can perform some operations on set(s). Let’s see some of them here.

Union of Sets

Union of two sets contains the elements of both the sets. That is, A ∪ B = {x | x ∈ A or x ∈ B}

It is denoted by the symbol ∪. So, A ∪ B means A union B.

The shaded portion in the following Venn diagram represents the union of two sets, A and B.

Properties of Union operation

Property 1: Commutative Law

The order of sets doesn’t matter, i.e. A ∪ B = B ∪ A

Property 2

(A ∪ B) ∪ C = A ∪ (B ∪ C)

Property 3: Distributive Law

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Property 4

• Union of a set with empty set is that set itself, i.e. A ∪ φ = A
• Union of a set with itself is that set itself, i.e. A ∪ A = A
• Union of a set with universal set is that universal set, i.e. A ∪ U = U

Intersection of Sets

Intersection of two sets contains the elements that are common in both the sets. That is, A ∩ B = {x | x ∈ A and x ∈ B}

It is denoted by the symbol ∩. So, A ∩ B means A intersection B.

The shaded portion in the following Venn diagram represents the intersection of two sets, A and B.

Properties of Intersection operation

Property 1: Commutative Law

The order of sets doesn’t matter, i.e. A ∩ B = B ∩ A

Property 2

(A ∩ B) ∩ C = A ∩ (B ∩ C)

Property 3: Distributive Law

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Property 4

• Intersection of a set with empty set is empty set, i.e. A ∩ φ = φ
• Intersection of a set with itself is that set itself, i.e. A ∩ A = A
• Intersection of a set with universal set is that set itself, i.e. A ∩ U = A

Difference of Sets

Difference of two sets is denoted by - sign. So, difference of sets A and B, will be represented as A - B.

A- B will have all the elements of set A, that are not in set B. That is, A - B = {x | x ∈ A and x ∉ B}

The shaded portion in the following Venn diagram represents A - B.

Complement of a Set

To represent complement of a set, we add ' infront of the name of the set. For example, complement of set A will be A’.

Let A be a set within the universal set U. Then A’ will have all the elements of the universal set, except those that are in set A. That is, A’ = {x | x ∈ U and x ∉ A}, or A′ = U – A

The shaded portion in the following Venn diagram represents the complement of set A, i.e. A’.

Properties of Complement of Sets

Property 1

Complement of universal set is empty set, and vice-versa. That is, U′ = φ and φ′ = U

Property 2

• A ∪ A′ = U
• A ∩ A′ = φ

Property 3: De Morgan’s law

The complement of the union of two sets = Intersection of their complements, i.e. (A ∪ B)′ = A′ ∩ B′

The complement of the intersection of two sets = Union of their complements, i.e. (A ∩ B)′ = A′ ∪ B′

Property 4: Law of Double Complementation

(A′)′ = A