Sets and Their Representation

Learn set notation, roster and set-builder forms, and standard number sets for JEE Mathematics.

5 min read

Introduction

A set is a well-defined collection of distinct objects called elements or members. The concept of sets was developed by Georg Cantor in the late 19th century and forms the basis of modern mathematics.

What Makes a Set “Well-Defined”?

A collection is well-defined if we can determine definitively whether any given object belongs to it or not.

Well-defined examples:

  • The set of vowels in the English alphabet
  • The set of prime numbers less than 20

Not well-defined examples:

  • The set of beautiful flowers (subjective)
  • The set of tall people (no clear criterion)
Notation

Sets are usually denoted by capital letters: $A, B, C, X, Y, Z$

Elements are denoted by lowercase letters: $a, b, c, x, y, z$

If $x$ is an element of set $A$, we write $x \in A$ ("$x$ belongs to $A$")

If $x$ is not in $A$, we write $x \notin A$

Representation of Sets

1. Roster (Tabular) Form

List all elements within curly braces, separated by commas.

Examples:

  • Vowels: $V = \{a, e, i, o, u\}$
  • First five primes: $P = \{2, 3, 5, 7, 11\}$
  • Even numbers from 2 to 10: $E = \{2, 4, 6, 8, 10\}$

Rules:

  • Order doesn’t matter: $\{1, 2, 3\} = \{3, 1, 2\}$
  • Repetition is ignored: $\{1, 2, 2, 3\} = \{1, 2, 3\}$

2. Set-Builder Form

Describe the property that elements satisfy.

$$A = \{x : \text{condition on } x\}$$

Read as: “A is the set of all $x$ such that [condition]”

Examples:

  • $A = \{x : x \in \mathbb{N}, x < 6\} = \{1, 2, 3, 4, 5\}$
  • $B = \{x : x^2 = 4\} = \{-2, 2\}$
  • $C = \{x : x \text{ is a vowel}\} = \{a, e, i, o, u\}$
When to Use Each Form
  • Roster form: When the set has few elements or follows a clear pattern
  • Set-builder form: When the set is infinite or describing a property is cleaner

Standard Number Sets

SymbolNameDescriptionExample Elements
$\mathbb{N}$Natural NumbersPositive counting numbers$1, 2, 3, 4, ...$
$\mathbb{W}$Whole NumbersNatural numbers with zero$0, 1, 2, 3, ...$
$\mathbb{Z}$IntegersWhole numbers and negatives$..., -2, -1, 0, 1, 2, ...$
$\mathbb{Z}^+$Positive IntegersSame as $\mathbb{N}$$1, 2, 3, ...$
$\mathbb{Q}$Rational NumbersFractions of integers$\frac{1}{2}, -\frac{3}{4}, 0.5$
$\mathbb{R}$Real NumbersAll points on number line$\pi, \sqrt{2}, -3.14$

Relationships Between Number Sets

$$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$
graph LR
    N[ℕ Natural] --> W[ℤ Whole]
    W --> Z[ℤ Integers]
    Z --> Q[ℚ Rationals]
    Q --> R[ℝ Reals]

Interactive Demo: Visualize Set Operations

Explore unions, intersections, and complements with Venn diagrams.

Types of Sets

1. Empty Set (Null Set)

A set with no elements, denoted by $\emptyset$ or $\{\}$.

Examples:

  • $\{x : x \in \mathbb{R}, x^2 = -1\} = \emptyset$ (no real solution)
  • $\{x : x \text{ is both even and odd}\} = \emptyset$
Common Mistake

$\emptyset \neq \{\emptyset\}$

  • $\emptyset$ is the empty set (has 0 elements)
  • $\{\emptyset\}$ is a set containing the empty set (has 1 element)

2. Singleton Set

A set with exactly one element.

Example: $A = \{5\}$ or $B = \{\sqrt{2}\}$

3. Finite and Infinite Sets

  • Finite Set: Has a countable number of elements

    • $A = \{1, 2, 3, 4, 5\}$ (finite, 5 elements)

  • Infinite Set: Has unlimited elements

    • $\mathbb{N} = \{1, 2, 3, ...\}$ (infinite)

4. Equal Sets

Two sets are equal if they have exactly the same elements.

$$A = B \iff (x \in A \Leftrightarrow x \in B)$$

Example: $\{1, 2, 3\} = \{3, 2, 1\}$ (order doesn’t matter)

5. Equivalent Sets

Two sets are equivalent if they have the same number of elements (same cardinality).

$\{a, b, c\}$ and $\{1, 2, 3\}$ are equivalent (both have 3 elements)

Subsets

Definition

Set $A$ is a subset of set $B$ (written $A \subseteq B$) if every element of $A$ is also in $B$.

$$A \subseteq B \iff (x \in A \Rightarrow x \in B)$$

Proper Subset

$A$ is a proper subset of $B$ (written $A \subset B$) if $A \subseteq B$ and $A \neq B$.

Important Facts

  1. Every set is a subset of itself: $A \subseteq A$
  2. Empty set is a subset of every set: $\emptyset \subseteq A$
  3. If $A \subseteq B$ and $B \subseteq A$, then $A = B$

Number of Subsets

If set $A$ has $n$ elements:

  • Number of subsets = $2^n$
  • Number of proper subsets = $2^n - 1$
  • Number of non-empty subsets = $2^n - 1$
  • Number of non-empty proper subsets = $2^n - 2$

Example: For $A = \{1, 2\}$ (n = 2): Subsets: $\emptyset, \{1\}, \{2\}, \{1, 2\}$ — that’s $2^2 = 4$ subsets

Universal Set

The universal set $U$ is the set containing all objects under consideration for a particular problem.

Example: When discussing vowels and consonants: $U = \{a, b, c, ..., z\}$ (all letters)

Interval Notation (for Real Numbers)

IntervalSet-BuilderNumber Line
$[a, b]$$\{x : a \leq x \leq b\}$Closed (includes endpoints)
$(a, b)$$\{x : a < x < b\}$Open (excludes endpoints)
$[a, b)$$\{x : a \leq x < b\}$Half-open
$(a, b]$$\{x : a < x \leq b\}$Half-open
$[a, \infty)$$\{x : x \geq a\}$Includes $a$, no upper bound
$(-\infty, b)$$\{x : x < b\}$No lower bound, excludes $b$

Practice Problems

  1. Write in roster form: $A = \{x : x^2 - 5x + 6 = 0\}$

  2. Write in set-builder form: $B = \{1, 4, 9, 16, 25\}$

  3. Find all subsets of $\{a, b\}$

  4. Is $\{1, 2\}$ equal to $\{1, 2, 2, 1\}$? Explain.

Quick Check
How many proper subsets does a set with 4 elements have?

Next: Set Operations →