What is a Set?
A set is a well-defined collection of distinct objects.
Example: A = \{ 1, 2, 3, 4, 5 \}

What is an element of a Set?

  • The objects in a set are called its elements.
    • So in case of the above Set A , the elements would be 1, 2, 3, 4 and 5 . We can say, 1 \in A, \ 2 \in A
  • Usually we denote Sets by CAPITAL LETTERs like A, \ B, \ C etc. while their elements are denoted in small letters like x, y, z
  • If x is an element of A , then we say x belongs to A and we represent it as x \in A
  • If x is not an element of A , then we say that x does not belong to A and we represent it as x \notin A

How to describe a Set?

  • Roaster Method or Tabular Form
    • In this form, we just list the elements
    • Example A = \{1, 2, 3, 4 \} or B = \{ a, b, c, d, e \}
  • Set- Builder Form or Rule Method or Description Method
    • In this method, we list the properties satisfied by all elements of the set

Example A = \{ x : x \in N, x< 5 \}

Some examples of Roster Form vs Set-builder Form

Roster Form Set-builder Form
1 \{ 1, 2, 3, 4, 5 \} \{ x | x \in N , x < 6 \}
2 \{ 2, 4, ,6 ,8 10 \} \{ x | x = 2n , n \in N, 1 \geq n \leq 5 \}
3 \{ 1, 4, 9, 16, 25, 36 \} \{ x | x = n^2 , n \in N, 1 \geq n \leq 6 \}

Sets of Numbers

  1. Natural Numbers (N)

N = \{1, 2, 3,4 ,5 6, 7, \cdots \}

  1. Integers (Z)

Z = \{ \cdots , -3, -2, -1, 0, 1, 2, 3, 4, \cdots \}

  1. Whole Numbers ( W )

W = \{0, 1, 2, 3 4, 5, 6 \cdots \}

  1. Rational Numbers (Q)

\{ p/q : p \in Z, q \in Z, q \neq 0 \}

Finite Sets & Infinite Sets

Finite Set: A set where the process of counting the elements of the set would surely come to an end is called finite set

  • Example: All natural numbers less than 50
  • All factors of the number 36

Infinite Set: A set that consists of uncountable number of distinct elements is called infinite set.

Example: Set containing all natural numbers \{ x | x \in N, x > 100 \}

Cardinal number of Finite Set

The number of distinct elements contained in a finite set A is called the cardinal number of A and is denoted by n(A)

  • Example A = \{1, 2, 3, 4 \} then n(A) = 4
  • A = \{ x | x \ is \ a \ letter \  in \ the \ word \ \ 'APPLE' \} . Therefore A = \{ A, P, L, E \} and n(A) = 4
  • A = \{ x | x \ is \ the \ factor \ of \ 36 \} , Therefore A = \{ 1, 2, 3, 4, 6, 9, 12, 18, 36 \} and n(A) = 9

Empty Set

  • A set containing no elements at all is called an empty set or a null set or a void set.
  • It is denoted by \phi  (phai)
  • In roster form you write \phi = \{ \}
  • Also n ( \phi ) = 0  
    • Examples: \{ x | x \in N, 3 <  x < 4 \} = \phi  
    • \{ x | x \ is \ an \ even \ prime \ number,  x > 5 \} = \phi

Non Empty Set

  • A set which has at least one element is called a non-empty set
    • Example: A = \{ 1, 2, 3 \} or B = \{ 1 \}

Singleton Set

  • A set containing exactly one element is called a singleton set
    • Example: A = \{ a \} or B = \{ 1 \}

Equal Sets

  • Two set A and B are said to be equal sets and written as A = B if every element of A is in B and every element of B is in A
    • Example A = \{ 1, 2, 3, 4 \} and B = \{ 4, 2, 3, 1 \}
  • It is not about the number of elements. It is the elements themselves.
  • If the sets are not equal, then we write as A \neq B

Equivalent Sets

  • Two finite sets A and B are said to be equivalent, written as A \longleftrightarrow B , if  n(A) = n(B) , that is they have the same number of elements.
    • Example: A = \{ a, e, i, o, u \} and B = \{ 1, 2, 3, 4, 5 \} , Therefore n(A) = 5 and n(B) = 5 therefore A \longleftrightarrow B
  • Note: Two equal sets are always equivalent but two equivalent sets need not be equal.

Subsets

  • If A and B are two sets given in such a way that every element of A is in B , then we say A is a subset of B and we write it as A \subseteq B
  • Therefore is A \subseteq B  and x \in A  then x \in B   
  • If A is a subset of B , we say B is a super set of A and is written as B \supseteq A
  • Every set is a subset of itself.
    • i.e. A \subseteq A , B \subseteq B  etc.
  • Empty set is a subset of every set
    • i.e. \phi \subseteq A, \ \ \phi \subseteq B
  • If A \subseteq B  and B \subseteq A , then A = B
  • Similarly, if A = B , then A \subseteq B  and B \subseteq A
  • If set A  contains n  elements, then there are 2^n  subsets of A

Power Set

  • The set of all possible subsets of a set A  is called the power set of A , denoted by P(A) . If A contains n elements, then P(A) = 2^2
    • i.e. if A = \{ 1, 2 \} , then P(A) = 2^2 = 4
    • Empty set is a subset of every set
    • So in this case the subsets are \{ 1 \}, \{ 2 \}, \{ 2, 3 \} \ \ \& \ \ \phi

Proper Subset

Let A be any set and let B be any non-empty subset. Then A  is called a proper subset of B , and is written as A \subset B , if and only if every element of A  is in B , and there exists at least one element in B  which is not there in A .

i.e. if A \subseteq B and A \neq B then A \subset B

Please note that \phi has no proper subset

A set containing n elements has (2n - 1) proper subsets.

i.e. if A = \{1, 2, 3, 4 \} , then the number of proper subsets is (24 - 1) = 15

Universal Set

If there are some sets in consideration, then there happens to be a set which is a super set of each one of the given sets. Such a set is known as universal set, to be denoted by U or \xi  .

i.e. if A = \{1, 2 \}, B = \{3, 4 \} , and C = \{1, 5 \} , then U or \xi = \{1, 2, 3, 4, 5 \}

Operations on Sets

Union of Sets

The union of sets A and B , denoted by A \cup B , is the set of all those elements, each one of which is either in A or in B or in both A and B

If there is a set A = \{2, 3 \} and B = \{a, b \} , then A \cup B = \{2, 3, a, b \}

So if A \cup B = \{x | x \in A \ or \ x \in B \} then x \in A \cup B which means x \in A or x \in B

And if x \notin A \cup B which means x \notin A or x \notin B

Interaction of Sets

The intersection of sets A and B is denoted by A \cap B , and is a set of all elements that are common in sets A and B .

  1. if A = \{1, 2, 3 \} and B = \{2, 4, 5 \} , then A \cap B = \{2 \} as 2 is the only common element.

Thus A \cap B = \{x : x \in A \ and \  x \in B \} then x \in A \cap B i.e. x \in A and x \in B

And if x \notin A \cap B i.e. x \notin A and x \notin B

Disjointed Sets

Two sets A and B are called disjointed, if they have no element in common. Therefore:

x \notin A \cap B i.e. x \notin A and x \notin B

Intersecting sets

Two sets are said to be intersecting or overlapping or joint sets, if they have at least one element in common.

Therefore two sets A and B are overlapping if and only if A \cap B \neq \phi

Intersection of sets is Commutative

i.e. A \cap B = B \cap A  for any sets A and B

Intersection of sets is Associative i.e. for any sets, A, B, C

(A \cap B) \cap C = A \cap ( B \cap C)

A \subseteq B , then A \cap B = A

Since A \subseteq  \xi , so A \cap \xi = A

For any sets A and B we have

A \cap B \subseteq A and A \cap B \subseteq B

A \cap  \phi = \phi for every set A

Difference of Sets

For any two sets A and B , the difference A - B is a set of all those elements of A which are not in B .

i.e. if A = \{1, 2, 3, 4, 5 \} and B = \{4, 5, 6 \} , Then A - B = \{1, 2, 3 \} and B - A = \{6 \}

Therefore A - B = \{ x | x \in A \ and \  x \notin B \} , then x \in A - B then x \in A but x \notin B

If A \subseteq B then A - B = \phi

Complement of a Set

Let x be the universal set and let A \subseteq x . The the complement of A , denoted by A' is the set if all those elements of x which are not in A .

i.e. let \xi = \{1, 2, 3, 4, 5, 6, 7, 8 \} and A= \{2, 3, 4 \} , then A'= \{1, 5, 6, 7, 8 \}

Thus A' = \{x | x \in \xi \ and \  x \notin A \} clearly x \in A'  and x \notin A

Please note

 \phi' =  \xi and \xi' = \phi

A \cup A' = \xi  and A \cap A' = \phi

Disruptive laws for Union and Intersection of Sets

For any three sets , B, C , we have the following

A \cup ( B \cap C) = (A \cup B) \cap (A \cup C)

Say A = \{1,2 \}, B = \{2, 3 \} and C = \{3, 4 \}

Therefore A \cup ( B \cap C) = \{1, 2, 3 \} and (A \cup B) \cap (A \cup C) = \{1, 2, 3 \} and hence equal

A \cap ( B \cup C) = (A \cap B) \cup (A \cap C)

Say A = \{1,2 \}, B = \{2, 3 \} and C = \{3, 4 \}

Then A \cap ( B \cup C) = \{2 \} and (A \cap B) \cup (A \cap C) = \{2 \} and hence equal

Disruptive laws for Union and Intersection of Sets

De-Morgan’s Laws

Let A and B be two subsets of a universal set \xi , then

(A \cup B)' = A' \cap B'

(A \cap B)' = A. \cup B'

Let \xi = \{1,2,3,4,5,6 \} and A = \{1,2,3 \} and B = \{3,4,5 \}

Then A \cup B = \{ 1, 2, 3, 4, 5 \} , therefore (A \cup B)' = \{ 6 \}

A' = \{4, 5, 6 \} and B'= \{1, 2, 6 \}

Therefore A' \cap B' = \{6 \} . Hence proven.