Sets

1. Introduction to Sets

A set is a well-defined collection of objects.

Objects, elements, and members of a set are synonymous terms.
Sets are usually denoted by capital letters $A, B, C, X, Y, Z$ , etc.
The elements of a set are represented by small letters $a, b, c, x, y, z$ , etc.

Notation:

If $a$ is an element of a set $A$ , we say that " $a$ belongs to $A$ " and write $a \in A$ .
If $b$ is not an element of a set $A$ , we write $b \notin A$ .

2. Representation of Sets

There are two methods of representing a set:

(A) Roster or Tabular Form

All elements are listed, separated by commas, and enclosed within braces $\{\}$ .

Example: The set of vowels in English alphabet is $\{a, e, i, o, u\}$ .
Note: Order of elements is immaterial. Elements are not generally repeated.

(B) Set-Builder Form

All the elements of a set possess a single common property which is not possessed by any element outside the set.

Example: $V = \{x : x \text{ is a vowel in English alphabet}\}$ .

3. Types of Sets

(A) Empty Set (Null/Void Set)

A set containing no elements. Denoted by $\phi$ or $\{\}$ .

Example: $\{x : x \in \mathbb{N}, 1 < x < 2\} = \phi$ .

(B) Finite and Infinite Sets

Finite: A set which is empty or consists of a definite number of elements.
Infinite: A set which is not finite (e.g., set of natural numbers $\mathbb{N}$ ).

(C) Equal Sets

Two sets $A$ and $B$ are equal if they have exactly the same elements. We write $A = B$ .

(D) Singleton set

Exactly one element.

Example: $\{0\}$

(E) Equivalent sets

$A$ and $B$ are equivalent if they have same number of elements, i.e., $n(A)=n(B)$ .

4. Subsets

A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$ . $A \subset B \iff a \in A \implies a \in B$

Intervals as subsets of $\mathbb{R}$ :
Open interval: $(a, b) = \{y : a < y < b\}$
Closed interval: $[a, b] = \{y : a \le y \le b\}$

Important facts:

$\varnothing\subseteq A$ for every set $A$ .
$A\subseteq A$ .
If $A\subseteq B$ and $B\subseteq A$ , then $A=B$ .

Power Set

The collection of all subsets of a set $A$ is called the Power Set of $A$ , denoted by $P(A)$ .

If $n(A) = m$ , then $n(P(A)) = 2^m$ .

5. Venn Diagrams

Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. The universal set $U$ is usually represented by a rectangle and its subsets by circles.

6. Operations on Sets

(A) Union of Sets

The union of two sets $A$ and $B$ is the set which consists of all the elements of $A$ and all the elements of $B$ , the common elements being taken only once. $A \cup B = \{x : x \in A \text{ or } x \in B\}$

(B) Intersection of Sets

The intersection of sets $A$ and $B$ is the set of all elements which are common to both $A$ and $B$ . $A \cap B = \{x : x \in A \text{ and } x \in B\}$

(C) Difference of Sets

The difference of sets $A$ and $B$ in this order is the set of elements which belong to $A$ but not to $B$ . $A - B = \{x : x \in A \text{ and } x \notin B\}$

(D) Symmetric Difference (often asked)

$A\Delta B = (A - B)\cup(B - A)$ Meaning: elements that are in exactly one of the sets.

7. Complement of a Set

Let $U$ be the universal set and $A$ a subset of $U$ . The complement of $A$ is the set of all elements of $U$ which are not the elements of $A$ . $A' = U - A = \{x : x \in U \text{ and } x \notin A\}$

Properties:

$(A')' = A$
$A\cup A' = U$
$A\cap A' = \varnothing$
$U' = \varnothing$ , $\varnothing' = U$

De Morgan's Laws

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

8. Practical Problems (Formulas)

If $A$ and $B$ are finite sets, then:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
If $A \cap B = \phi$ , then $n(A \cup B) = n(A) + n(B)$

Example 1

Write the solution set of the equation $x^2 + x - 2 = 0$ in roster form.

Solution: $(x-1)(x+2)=0 \implies x = 1, -2$ . Set $A = \{1, -2\}$ .

Example 2

Write the set $A = \{1, 4, 9, 16, 25, \dots\}$ in set-builder form.

Solution: $A = \{x : x = n^2, n \in \mathbb{N}\}$ .

Example 3

Let $A = \{1, 2, 3, 4, 5, 6\}$ . Insert the appropriate symbol $\in$ or $\notin$ in the blank: $5 \dots A$ .

Solution: Since 5 is in the list, $5 \in A$ .

Example 4

Which of the following is an empty set? $A = \{x : x \text{ is an even prime number greater than 2}\}$ .

Solution: The only even prime number is 2. There is no even prime $> 2$ . Hence $A = \phi$ . It is an empty set.

Example 5

Let $A = \{a, b\}$ . List all subsets of $A$ .

Solution: Subsets are: $\phi, \{a\}, \{b\}, \{a, b\}$ .

Example 6

If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$ , find $A \cup B$ .

Solution: Combine all elements: $A \cup B = \{1, 2, 3, 4, 5\}$ .

Example 7

From the above example, find $A \cap B$ .

Solution: Common element is 3. $A \cap B = \{3\}$ .

Example 8

Let $U = \{1, 2, 3, 4, 5, 6\}$ , $A = \{2, 3\}$ . Find $A'$ .

Solution: $A' = U - A = \{1, 4, 5, 6\}$ .

Example 9

If $X$ and $Y$ are two sets such that $X$ has 40 elements, $X \cup Y$ has 60 elements and $X \cap Y$ has 10 elements, how many elements does $Y$ have?

Solution: Using $n(X \cup Y) = n(X) + n(Y) - n(X \cap Y)$ : $60 = 40 + n(Y) - 10$ $60 = 30 + n(Y)$ $n(Y) = 30$ .

Example 10

In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Solution: Let $H$ = Hindi speakers, $E$ = English speakers. $n(H \cup E) = 400$ , $n(H) = 250$ , $n(E) = 200$ . $400 = 250 + 200 - n(H \cap E)$ $400 = 450 - n(H \cap E)$ $n(H \cap E) = 50$ .

Example 11

Find the power set of $A = \{1\}$ .

Solution: Subsets are $\phi$ and $\{1\}$ . $P(A) = \{\phi, \{1\}\}$ .

Example 12

State true or false: $\{a, b\} \subset \{b, c, a\}$ .

Solution: Both elements $a$ and $b$ are in the second set. True.

Questions and Answers (Board Exam)

Q1. Define a Set.

Answer: A set is a well-defined collection of objects. "Well-defined" means we can definitely decide whether a given object belongs to the collection or not.

Q2. What is a Power Set?

Answer: The collection of all subsets of a set $A$ is called the power set of $A$ . It is denoted by $P(A)$ . In $P(A)$ , every element is a set.

Q3. State De Morgan's Laws.

Answer: For any two sets $A$ and $B$ :

The complement of the union is the intersection of the complements: $(A \cup B)' = A' \cap B'$ .
The complement of the intersection is the union of the complements: $(A \cap B)' = A' \cup B'$ .

Q4. What is the difference between $\{0\}$ and $\phi$ ?

Answer: $\{0\}$ is a singleton set containing one element (the number 0). $\phi$ is an empty set containing no elements.

Q5. If $A \subset B$ , what is $A \cup B$ ?

Answer: If $A$ is contained in $B$ , their union is the larger set $B$ .