Sets and its types

Defining Sets

Example: The collection of vowels in English alphabets. The set contains five elements, namely, a, e, i, o, u.

Forms of Describing a Set

Sets can be described in two forms:

(i) Roster Form or Tabular Form

In this form, a set is described by listing elements, separated by commas, within braces {}. In roster method, φ is denoted by { }.

Eg: The set of vowels of the English Alphabet may be described as {a, e, i, o, u}

Eg: {x ∈ R : x² = -2} = φ, where R is positive real numbers.

• The order in which the elements are written in a set makes no difference.
• Repetition of an element has no effect.

(ii) Set-Builder Form

In this form, a set is described by a characterizing property p(x) of its elements x. In such a case, the set is described as {x : p(x) holds}.

Eg: The set A = {1, 2, 3, 4, 5, 6, 7, 8} can be written as A = {x ∈ N : x ≤ 8}

Types of Sets

(i) Empty Set

A set is said to be empty or null if it has no element, and it is denoted by ∅.

(ii) Singleton Set

A set consisting of a single element is called a singleton set.

The set {5} is a singleton set.

(iii) Finite Set

A set is called finite if its elements can be listed by natural numbers 1, 2, 3, … n, and the process of listing terminates at a certain natural number, say n.

(iv) Infinite Set

A set whose elements cannot be listed by the natural numbers 1, 2, 3, … n, for any natural number ‘n’, is called an infinite set.

Eg:

• Set of all points on a plane
• Set of all lines
• {x ∈ R : 0 < x < 1}

(v) Equivalent Set

Two finite sets A and B are equivalent if their cardinal numbers are the same, i.e., n(A) = n(B).

(vi) 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.

Subsets

Definition of a Subset

Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B.

Notation: If A is a subset of B, then A ⊆ B, or A is contained in B.

If A is a subset of B, we say that B contains A, or B is a superset of A, and we write B ⊇ A.

If A is not a subset of B, we write A ⊈ B.

Improper Subsets

As every set is a subset of itself and the empty set is the subset of every set, these two subsets are called improper subsets.

Intervals as Subsets of Real Numbers

Closed Interval

Let a and b be two given real numbers such that a < b. Then, the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by [a, b].

Diagram:

Open Interval

If a and b are two real numbers such that a < b, then the set of all real numbers x satisfying a < x < b is called an open interval and is denoted by (a, b) or ]a, b[.

Diagram:

In this set, a and b are not included.

Semi-Open or Semi-Closed Interval

If a and b are two real numbers such that a < b, then the sets (a, b] = {x ∈ R : a < x ≤ b} and [a, b) = {x ∈ R : a ≤ x < b} are known as semi-open or semi-closed intervals.

Universal Set

A set that contains all sets in a given context is called the universal set.

Eg: A = {1, 2, 3}, B = {2, 4, 6}

U = {1, 2, 3, 4, 5, 6}

Power Set

Let A be a set. Then the collection or family of all subsets of A is called the power set of A and is denoted by P(A).

Eg: Let A = {1, 2, 3}

Then, the subsets of A are

φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and {1, 2, 3}

So P(A) = { φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

Venn Diagrams

Union of Sets

Let A and B be two sets. The union of A and B is the set of all those elements which belong to either A or B or both A and B.

Then, A ∪ B = { x : x ∈ A or x ∈ B }

Diagram:

Intersection of Sets

Let A and B be two sets. The intersection of A and B is the set of all those elements that belong to both A and B.

So A ∩ B = { x : x ∈ A and x ∈ B }

Diagram:

Disjoint Sets

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

Diagram:

Difference of Sets

Let A and B be two sets. The difference of A and B, written as A – B, is the set of all those elements of A which do not belong to B.

Then, A – B = { x : x ∈ A and x ∉ B }

Diagram:

Symmetric Difference of Sets

Let A and B be two sets. The symmetric difference of sets A and B is the set (A – B) ∪ (B – A) and is denoted by A ∆ B.

Then, A ∆ B = (A – B) ∪ (B – A) = { x : x ∈ A ∆ B }

Diagram:

Complement Set

Let U be the universal set and let A be a set such that A ⊆ U. Then the complement of A with respect to U is denoted by A’ or A^c or U – A, and is defined as the set of all those elements of U which are not in A.

Then A^c = { x ∈ U : x ∉ A }

Diagram: