Sets

A set is a collection of well-defined objects that share some common property. It can be a group of any items, such as the names of the months in a year, the days in a week, or a list of variables or constants. Sets are named and represented in capital letters.

Here are some examples of sets:

A = {-5, -3, -1, 1, 3, 5}

B = {2, 3, 5, 7, 11, 13, …}

Standard Sets

Here are some standard sets in mathematics:

• Set of natural numbers; ℕ = {1, 2, 3, …}
• Set of whole numbers; 𝕎 = {0, 1, 2, 3, …}
• Set of integers; ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}
• Set of rational numbers; ℚ = {${\dfrac{x}{y}}$ | ‘x’ and ‘y’ are integers and y ≠ 0}
• Set of irrational numbers; ℚ’ = {x is not rational}
• Set of real numbers; ℝ = ℚ ∪ ℚ’

Writing 

In a set, we separate each term by a comma and then put curly brackets ‘{ }’ around the whole group. These curly brackets ‘{ }’ are called ‘set brackets’ or ‘braces.’

Elements

The terms or items present in the set are called elements or members of that set. The elements are represented by lowercase letters.

The first set has 6 terms, but the second set has an uncountable number of terms. The three dots together are called ellipses, ‘…’ which means ‘continue on.’ 

Sometimes, the ellipsis is used in the middle of terms to avoid writing many terms. For example, in the set C = {a, b, c, …, z}, there are 26 letters, and thus there are 26 terms in set ‘C.’

If ‘A’ is a set containing elements ‘a’ and ‘b,’ A = {a, b}. 

Here, we also use the symbol ‘∈’ (read as ‘belongs to’) to represent the element ‘a’ in  the set ‘A.’

 Since ‘a’ is in the set ‘A,’ but another element ‘c’ is not in the set ‘A,’ we can write a ∈ A and c ∉ A, read as ‘a belongs to A’ and ‘c does not belong to A.’ 

Cardinality

The number of elements present in a set is called the cardinal number, cardinality, or order of a set. 

For example, the cardinality of the set ‘A’ of natural numbers A = {1, 2, 3, 4} is 4, which is written as n(A) = |A| = 4

Representing in Set Theory

There are some different set notations used to represent sets.  

Semantic or Statement Form

Semantic notation describes a statement to represent the elements of a set.

For example, a set A = {0, 1, 2, 3, 4, 5, 6, 7, 8} in semantic form is written as {a set of whole numbers less than 8}

Roster Form

This is the most common way of representing sets. In roster notation, the elements are enclosed within curly brackets and are separated by commas.

For example, the set of whole numbers less than 8 is represented in roster form as B = {0, 1, 2, 3, 4, 5, 6, 7, 8}

Set Builder Form

In set-builder notation, a rule or statement describes the common property of all the elements. 

For example, a set of whole numbers less than 8 is represented as B = {x | x is a whole number and x < 8}

Types

Empty Set (Null Set)

A set that does not contain any element is called an empty set or null set. It is represented using the symbol ‘ɸ,’ read as ‘phi.’ 

For example, Set A = { }

Singleton Set (Unit Set)

A set that contains only one element is called a singleton or unit set. 

For example, Set A = {0} and Set B = {9}

Finite Set

A set that contains a finite or countable number of elements is called a finite set. 

For example, Set B = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}

Infinite Set

A set containing an infinite number of elements is called an infinite set.

For example, the set of negative integers is {…, -4, -3, -2, -1}

Equivalent Set

If two sets contain the same number of elements, even if the elements are different, they are called equivalent sets. 

For example, if A = {-8, -4, -2, 0} and B = {p, q, r, s}, since n(A) = n(B), sets A and B are equivalent.

Equal Set

If two sets contain the same elements, they are called equal sets. 

For example, if A = {4, 5, 6}, B = {4, 5, 6}, and C = {5, 6, 4}, then sets A, B, and C are equal (A = B = C) irrespective of the order of the elements. 

Unequal Set

Two sets are said to be unequal if they have at least one unique element. 

For example, if A = {4, 5, 6} and B = {2, 3, 4}, then set A and B are unequal, which is written as A ≠ B.

Overlapping Set

If A and B are two sets and at least one element from A is present in B or vice versa, then the two sets A and B are said to be overlapping. 

For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6, 7}, then elements 3 and 4 are common in both sets A and B. Thus, A and B are overlapping sets.

Disjoint Set

If A and B are two sets with no common elements in between, then sets A and B are called disjoint sets. 

For example, if A = {m, n, s, t} and B = {5, 6, 7, 8, 9}, then set A and set B have no common elements in between. They are disjoint sets.

Subset and Superset

If A and B are two sets such that every element in A is also present in B, then set A is the subset of set B, represented by A ⊆ B. Here, B is known as the superset of set A, which is represented as B ⊇ A.

For example, if A = {2, 3, 5, 7, 11} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, since all the elements of A are present in B, A is a subset of set B, written as A ⊆ B.  Also, set B is the superset of set A (B ⊇ A.)

Again, if X = {1, 2, 3, 4}, Y = {1, 2, 3, 4}, and Z = {1, 2, 3, 4, 5}, then every element of X is present in Y, whereas element 5 of the set Z is not in set X. Thus, set X is a proper subset of set Z.

In general, A is a proper subset of B if and only if every element of A is also in B, but there is at least one element in B that is not present in A. It is represented as A ⊂ B.

Power Set

A power set is a set of all subsets that the set could contain.

For example, if A = {0, 1}, the power set of A is represented by P(A). Here, P(A) = {ɸ, {0}, {1}, {0, 1}}

Universal Set

A universal set, represented by the letter ‘U,’ is the collection of all the elements relevant to a certain condition. 

For example, if U is the set of real numbers, the sets of natural numbers and rational numbers are the subsets of this universal set.

Here is the list of the different types of sets we learned.

Symbols

Here is the list of all the symbols used to define the elements in a set.

Set Operations

Union

The union of sets, represented by A ∪ B, lists all the elements in sets A and B or the elements present in both sets A and B. 

For example, if A = {4, 5} and B = {5, 6, 7}, then A ∪ B = {4, 5} ∪ {5, 6, 7} = {4, 5, 6, 7}

Intersection

The intersection of sets, represented by A ∩ B, lists the common elements of sets A and B. 

For example, if A = {4, 5} and B = {5, 6, 7}, then A ∩ B = {4, 5} ∩ {5, 6, 7} = {5}

Difference

The set difference represents the elements in set A that are not present in set B. It is represented by A – B. 

For example, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. A – B = {1, 2}

Complement

The complement of a set A, denoted by A’, represents all elements in the universal set that are absent in set A. Also, A’ is represented as U – A, the difference between the elements of the universal set and set A.

For example, if the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and set A = {2, 3, 4, 5, 6}, then A’ = {1, 7, 8, 9, 10}

Cartesian Product

The cartesian product of two sets, A and B, written as A × B, is the product of two non-empty sets. Here, the ordered pairs of elements are obtained, which means the first element is obtained from the first set and the second element from the second set.

For example, {5, 6} × {7, 8} = {(5, 7), (6, 7), (5, 8), (6, 8)}

Symmetric Difference

The symmetric difference between two sets, A and B, represented by A Δ B, represents the remaining elements after removing the common elements from the union of the two sets. 

Mathematically, it is written as 

A Δ B = (A ∪ B) – (A ∩ B) 

⇒ A Δ B = (A – B) ∪ (B – A), which represents the elements of only set A and only set B.

Here is a summary of the commonly used set operations:

Formulas

If ‘A’ and ‘B’ are two overlapping sets, then

n(A ∪ B)	n(A) + n(B) – n(A ∩ B)
n(A ∩ B)	n(A) + n(B) – n(A ∪ B)
n(A)	n(A ∪ B) + n(A ∩ B) – n(B)
n(B)	n(A ∪ B) + n(A ∩ B) – n(A)
n(A – B)	n(A ∪ B) – n(B)
n(A – B)	n(A) – n(A ∩ B)
n(A ∪ B ∪ C)	n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)

If ‘A’ and ‘B’ are two disjoint sets, then

n(A ∪ B)	n(A) + n(B)
(A ∩ B)	ɸ
n(A – B)	n(A)

Properties

If ‘A,’ ‘B,’ and ‘C’ are three sets, they follow the following properties.

Commutative Property	A ∪ B = B ∪ AA ∩ B = B ∩ A
Associative property	(A ∪ B) ∪ C = A ∪ (B ∪ C)(A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive Property	A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Identity Property	A ∪ ɸ = AA ∩ ∪ = A
Complement Property	A ∪ A’ = ∪
Idempotent Property	A ∪ A = AA ∩ A = A

Representing with Venn Diagram

Sets are often represented pictorially using a Venn diagram for ease of understanding. It also illustrates how the given sets relate to each other.

In a Venn diagram, the elements of each set are enclosed in a circle. Sometimes, a rectangle surrounds the circles, representing the universal set.

Here are the commonly used set operations with their Venn diagrams:

The last two operations are known as De Morgan’s Laws. For more, follow De Morgan’s Law.

Solved Examples