Set Theory – Concepts and Definitions

What is a Set?

A set is a group or collection of well-defined and distinct objects. These objects are called the elements or members of the set.

Set theory was developed by Georg Cantor, a German mathematician.

• A set is usually named using capital letters: A, B, C…

• Elements are written in lowercase letters inside curly brackets: {a, b, c}

If x is in set A, we write: x ∈ A

If y is not in set A, we write: y ∉ A

Example:

Let A = {a, e, i, o, u} → the set of vowels in English.

Ways to Represent a Set

1. Roster (Tabular or Listing) Method:

This is a method of describing a set by listing each element of the set inside the symbol { }. In listing the elements of the set, each distinct elements is listed once and the order of the elements does not matter.

Example: Colors of a rainbow:

A = {red, orange, yellow, green, blue, indigo, violet}

2. Set-Builder (Rule) Method:

It is a method that lists the rules that determine whether an object is an element of the set rather than the actual elements.

Example:

A = {x | x is a city in India}

(“A is the set of all x such that x is a city in India.”)

Verbal Descriptions of Sets

1. A = {1, 3, 5, 7, …} → The set of odd numbers

2. B = {a, b, c, …, z} → The set of lowercase English letters

3. C = {4, 8, 12, …, 96} → Multiples of 4 between 0 and 100

List Elements Using the Roster Method

1. A = {x | x > 7, x is an odd number} → A = {9, 11, 13, 15, …}

2. B = {x | 7 < x < 8, x is a real number} → Infinite values between 7 and 8

3. C = {x | x is a city in Meghalaya} → C = {Shillong, Jowai, Tura, …}

4. D = {x | x is a whole number between 7 and 10} → D = {8, 9}

Write Rules Using Set-Builder Method

1. A = {a, e, i, o, u} → A = {x | x is a vowel in English alphabet}

2. B = {3, 6, 9, …, 30} → B = {x | 3 ≤ x ≤ 30, x is a multiple of 3}

3. C = {Delhi, Mumbai, Kolkata, Chennai, Patna, Jaipur, Ranchi} → C = {x | x is a major city in India}

Common Sets in Math

N = Natural Numbers

Z = Integers 

Q = Rational Numbers

R = Real Numbers

Z⁺ = Positive Integers

Q⁺ = Positive Rational Numbers

R⁺ = Positive Real Numbers

Types of Sets

1. Empty Set (Null Set): A set which does not contain any element is called empty set or null set or void set denoted by the symbol { } or ϕ.

Example: A = {x | 1 < x < 2, x is a whole number} → ∅

2. Singleton Set: A set consisting of a single element is called singleton set.

Example: B = {3}

3. Finite Set: A set is called a finite set if it is either void set or its elements can be listed counted by natural numbers 1, 2, 3, …. And the process of listing terminates at certain natural number n.

Example: C = {x | x is a multiple of 5 less than 100}

4. Infinite Set: A set whose elements cannot be listed or counted by the natural numbers is called an infinite set.

Example: D = {x | x is a real number greater than 1}

5. Equal Sets: Two sets A and B are said to be equal if every element of A is a member of B and every element of B is a member of A. If sets A and B are equal, we write A = B and A ≠ B when A and B are not equal.

Example: A = {1, 2}, B = {2, 1} → A = B

6. Equivalent Sets: Equivalent sets are sets with equal number of elements. These sets do not need to have the same exact elements.

Example: A = {a, b, c}, B = {1, 2, 3} → A and B are equivalent

Subsets and Supersets

Subset (⊆): If A and B are sets and every element of A is also an element of B, then A is a subset of (or is included in) B, denoted by A ⊆ B or equivalently B is a superset of (or includes) A, denoted by B ⊇ A.

• A ⊆ B if all elements of A are in B

Proper Subset (⊂): A ⊂ B if A is a subset of B but not equal to B

If A = {2, 3, 4} and B = {2, 3, 4, 5} are two sets, then “A is a proper subset of B” if all the members of A are also members of B, but in addition there exists at least one element 5 such that 5 ∈ B but 5 ∉ A. The notation for subset is very similar to the notation for “less than,” and means, in terms of the sets, “included in but not equal to.” Example:

If A = {2, 3, 4} and B = {2, 3, 4, 5}, then A ⸦ B (A is a proper subset of B)

Power of a Set (Power Set)

Power Set (P(A)): Set of all subsets of A

The set of all subset of a given set is called power set of that set. The collection of all subsets of a set A is called the power set of A denoted by P(A). In P(A), every element is a set. 

Example: A = {2, 4} → P(A) = {∅, {2}, {4}, {2, 4}}

Universal Set (U)

The universal set is a set that contains all sets under consideration, i.e., it is a superset of each of the given sets. Such a set is called universal set and is denoted by U.

Example: U = {all real numbers}

Venn Diagrams

Used to represent sets using shapes like circles inside rectangles.

• Rectangle → Universal set (U)

• Circles → Different sets (A, B, C)

A Venn diagram is a pictorial representation of sets. Venn diagrams were introduced in 1880 by John Venn. The universal set U is represented as a rectangle. Other sets are represented as circles. For example, 

If A = {1,2,3,4,5,6,7,8,9,10}

B = {1,2,3,4,5}

C ={4, 5,6,7}

Then, this situation is represented as:

Set Operations

1. Union (A ∪ B): All elements in A or B

2. Intersection (A ∩ B): Elements common to both A and B

3. Difference (B − A): Elements in B not in A

4. Complement (A'): Elements in U not in A

Let A and B be subsets of the universal set U. 

1. Union of Sets: The Union of A and B, denoted A ∪ B, is the set of all elements x in U such

that x is either in A or in B.

A ∪ B = {x ∈ U | x ∈ A or x ∈ B}

2. Intersection of Sets: The Intersection of A and B, denoted A ∩ B, is the set of all elements x

in U such that x is in both A and B. 

A ∩ B = {x ∈ U | x ∈ A and x ∈ B}

3. Difference of Sets: The Difference of B minus A, denoted B − A, is the set of all elements x in U such that x is in B, but not in A. 

B − A or B\A= {x ∈ U | x ∈ B and x ∉ A}

4. Complement of Sets: The complement of a set A, denoted as is the set of all elements  x in U such that  x is not in A. 

A' or Aᶜ  = {x ∈ U | x ∉ A }

Laws of Set Theory

1. Associative Law:

The associative law for the union and intersection of sets states that:

For Union: (A ∪ B) ∪ C=A ∪ (B ∪ C) 

For Intersection: (A ∩ B) ∩C=A ∩ (B ∩ C)

2.  Distributive Law:

The distributive law for the union and intersection of sets states that:

For Union: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 

For Intersection: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

3. Commutative Law:

For Union: A ∪ B = B ∪ A

For Intersection: A ∩ B = B ∩ A

4. De Morgan’s Law:

For Union: (A ∪ B)' = A' ∩ B'

For Intersection: (A ∩ B)' = A' ∪ B'

5. Idempotent Law:

For Union: A ∪ A = A

For Intersection: A ∩ A = A

6. Double Complement Law (Involution):

(A')' = A

7. Cardinal Rule:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Cartesian Product

The Cartesian product of sets A and B is the set of all ordered pairs:

A × B = {(a, b) | a ∈ A and b ∈ B}

Example:

A = {1, 2}, B = {3, 4}

A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}

For three sets:

A × B × C = {(a, b, c) | a ∈ A, b ∈ B, c ∈ C}