Set theory notes

 

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

Representation of Sets : There are two methods of representing a set -

• Roster or Tabular form : In the roster form, we list all the members of the set within braces { } and separate by commas.
• Set-builder form : In the set-builder form, we list the property or properties satisfied by all the elements of the sets.

Types of Sets

Empty Sets : A set which does not contain any element is called an empty set or the void set or null set and it is denoted by {} or Φ.

Singleton Set : A set consists of a single element, is called a singleton set.

Finite and infinite Set : A set which consists of a finite number of elements, is called a finite set, otherwise the set is called an infinite set.

Equal Sets : Two sets A and 6 are said to be equal, if every element of A is also an element of B or vice-versa, i.e. two equal sets will have exactly the same element.

Equivalent Sets : Two finite sets A and 6 are said to be equal if the number of elements are equal, 

                                i.e. n(A) = n(B)
Subset  : A set A is said to be a subset of set B if every element of set A belongs to set B. In symbols, we write A ⊆ B, if x ∈ A ⇒ x ∈ B

Power Set : The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). If the number of elements in A i.e. n(A) = n, then the number of elements in P(A) = 2n.

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

Venn-Diagrams : Venn diagrams are the diagrams, which represent the relationship between sets. In Venn-diagrams the universal set U is represented by point within a rectangle and its subsets are represented by points in closed curves within the rectangle.

Operations of Sets -

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}.

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}

Disjoint sets: Two sets Aand Bare said to be disjoint, if A ∩ B = Φ.

Intersecting or Overlapping sets : Two sets A and B are said to be intersecting or overlapping 

                                                           if A ∩ B ≠ Φ
Difference of sets : For any sets A and B, their difference (A – B) is defined as a set of elements, which belong to A but not to B. Thus, 

                                   A – B = {x : x ∈ A and x ∉ B}
also,                            B – A = {x : x ∈ B and x ∉ A}

Complement of a set : Let U be the universal set and A is a subset of U. Then, the complement of A is the set of all elements of U which are not the element of A. Thus, 

                                     A’ = U – A = {x : x ∈ U and x ∉ A}

Some Properties of Complement of Sets

A ∪ A’ = ∪
A ∩ A’ = Φ
∪’ = Φ
Φ’ = ∪
(A’)’ = A

Symmetric difference of two sets : For any set A and B, their symmetric difference (A – B) ∪ (B – A); (A – B) ∪ (B – A) defined as set of elements which do not belong to both A and B. It is denoted by A∆B. Thus, A ∆ B = (A – B) ∪ (B – A) = {x : x ∉ A ∩ B}.

Laws of Algebra of Sets

Idempotent Laws : For any set A, we have

A ∪ A = A

A ∩ A = A

Identity Laws: For any set A, we have

A ∪ Φ = A

A ∩ U = A

Commutative Laws : For any two sets A and B, we have

A ∪ B = B ∪ A

A ∩ B = B ∩ A

Associative Laws : For any three sets A, B and C, we have

A ∪ (B ∪ C) = (A ∪ B) ∪ C

A ∩ (B ∩ C) = (A ∩ B) ∩ C

Distributive Laws : If A, B and Care three sets, then

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De-Morgan’s Laws : If A and B are two sets, then

(A ∪ B)’ = A’ ∩ B’

(A ∩ B)’ = A’ ∪ B’

Formulae 

Union and Intersection of Two Sets : Let A, B and C be any three finite sets, then

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

If (A ∩ B) = Φ, then n (A ∪ B) = n(A) + 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(A ∩ C)+n(A ∩ B ∩ C)

Note :

• Every set is o subset of itself.
• The empty set is a subset of every set.
• The total number of subsets of a finite set containing n elements is 2n.
• Intervals as Subsets of R
• Let a and b be two given real numbers such that a < b, then

•  an open interval denoted by (a, b) is the set of real numbers

         {x : a < x < b}. 

• a closed interval denoted by [a, b] is the set of real numbers 

         {x : a ≤ x ≤ b}.

• Intervals closed at one end and open at the others are known as semi-open or semi-closed interval and denoted by (a, b] is the set of real numbers {x : a < x ≤ b} or [a, b) is the set of real numbers {x : a ≤ x < b}.