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

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, $1 \in A, \ 2 \in A$
• $\text{Usually we denote Sets by CAPITAL LETTERs like } A, \ B, \ C \text{ etc. while their }$ 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
• $\text{Example } A = \{1, 2, 3, 4 \} \text{ or } B = \{ a, b, c, d, e \}$
• Set- Builder Form or Rule Method or Description Method
• In this method, we list the properties satisfied by all elements of the set

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

Some examples of Roster Form vs Set-builder Form

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

Sets of Numbers

$\text{Natural Numbers } (N)$

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

$\text{Integers } (Z)$

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

$\text{Whole Numbers } ( W )$

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

$\text{Rational Numbers } (Q)$

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

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 $n(A)$

$\text{Example } A = \{1, 2, 3, 4 \} \text{ then } n(A) = 4$

$A = \{ x | x \text{ is a letter in the word 'APPLE'} \} \text{ Therefore } A = \{ A, P, L, E \} \\ \text{ and } n(A) = 4$

$A = \{ x | x \text{ is a factor of } 36 \} , \text{ Therefore } A = \{ 1, 2, 3, 4, 6, 9, 12, 18, 36 \} \\ \text{ and } n(A) = 9$

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: $\{ x | x \in N, 3 < x < 4 \} = \phi$
• $\{ x | x \text{ is an even prime number,} x > 5 \} = \phi$

Non Empty Set

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

Singleton Set

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

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 $A = \{ 1, 2, 3, 4 \} \text{ and } B = \{ 4, 2, 3, 1 \}$
• 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: $A = \{ a, e, i, o, u \} \text{ and } B = \{ 1, 2, 3, 4, 5 \}$, Therefore $n(A) = 5 \text{ and } n(B) = 5$ 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.

$\text{i.e. if } A = \{1, 2, 3, 4 \}$, 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 $U \text{ or } \xi$ .

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

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 $A \text{ or in } B$ or in both $A \text{ and } B$

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

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 $x \notin A$

$\phi' = \xi$ 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.