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 ⊂ 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 ⊆ B and  A ≠ B then A⊂ B
• Please note that ϕ 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 ξ .
• i.e. if A = {1, 2}, B = {3, 4}, and C = {1, 5}, then U or ξ = {1, 2, 3, 4, 5}

Operations on Sets

Union of Sets

• The union of sets A and B, denoted by A∪ 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∪ B = {2, 3, a, b}
• So if A∪ B = {x | x ∈ A or x ∈ B}  then  x ∈ A ∪ B  which means x ∈ A or x ∈ B
• And if x  ∉ A ∪ B  which means x ∉ A or x ∉ B

Interaction of Sets

• The intersection of sets A and B is denoted by A ∩ B  , and is a set of all elements that are common in sets A and B.
• e. if A = {1, 2, 3} and B = {2, 4, 5}, then A ∩ B = {2}  as 2 is the only common element.
• Thus A ∩ B = {x : x ∈ A and x ∈ B} then x ∈ A ∩ B i.e. x ∈ A and x ∈ B
• And if x ∉ A ∩ B  i.e. x ∉ A and x ∉ B

Disjointed Sets

• Two sets A and B are called disjointed, if they have no element in common. Therefore:
• x ∉ A ∩ B  i.e. x ∉ A and x ∉ 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 ∩ B ≠ ϕ
• Intersection of sets is Commutative
• i.e. A ∩ B  = B ∩ A for any sets A and B
• Intersection of sets is Associative i.e. for any sets, A, B,  C,
• (A ∩ B) ∩ C = A ∩ ( B ∩ C)
• A ⊆ B, then A ∩ B = A
• Since A ⊆  ξ, so A ∩ ξ = A
• For any sets A and B we have
•  A ∩ B ⊆ A and A ∩ B ⊆ B
• A ∩  ϕ = ϕ 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 ∈ A and x ∉ B}, then x ∈ A – B then x ∈ A but x ∉ B

• If A ⊆ B then A – B = ϕ

Complement of a Set

Let x be the universal set and let A ⊆ 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 ξ = {1, 2, 3, 4, 5, 6, 7, 8} and A= {2, 3, 4}, then A’={1, 5, 6, 7, 8}
• Thus A’={x | x ∈ ξ and x ∉ A} clearly x ∈A’ and x ∉ A

• ϕ’ =  ξ and ξ’ = ϕ
• A ∪ A’ = ξ and A ∩ A’ = ϕ

Disruptive laws for Union and Intersection of Sets

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

•   A ∪ ( B ∩ C) = (A ∪ B) ∩ (A ∪ C)

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

Therefore  A ∪ ( B ∩ C) = {1, 2, 3} and (A ∪ B) ∩ (A ∪ C) = {1, 2, 3} and hence equal

• A ∩ ( B ∪ C) = (A ∩ B) ∪ (A ∩ C)

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

Then A ∩ ( B ∪ C) = {2} and (A ∩ B) ∪ (A ∩ 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 ξ, then

• (A ∪ B)’ = A’ ∩ B’
• (A ∩ B)’ = A’∪ B’

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

Then A ∪ B = { 1, 2, 3, 4, 5}, therefore (A ∪ B)’ = {6}

A’ = {4, 5, 6} and B’={1, 2, 6}

Therefore A’ ∩ B’ = {6}. Hence proven.