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Sets

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Part of NCERT 11 Mathematics

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prerequisite

Ready Check

Make sure you're ready to start

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Are You Ready? Quick Check!
I understand what a 'collection' or 'group' means
I can tell if something belongs to a group or not
I know basic number types (even, odd, prime)
I understand inequalities (>, <, ≥, ≤)
I can solve simple equations like x² = 4
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overview

Why Sets Matter

See sets in your world

Sets Run Your Digital Life

Right now, sets are working behind the scenes:

Your WhatsApp groups? Each is a set of people.

Your Spotify 'Liked Songs'? A set of tracks.

Netflix recommendations? Sets of movies you might like.

Google search results? A set of matching web pages.

Every app, every database, every algorithm uses sets.

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concept

Well-Defined Collections

What makes a set a set?

The Golden Rule

For ANY collection to be a set, this must be true:

Can you clearly decide if something belongs? YES or NO?

No 'maybe'. No 'depends'. No opinions.

Everyone must agree on what's in and what's out.

This is called being well-defined.

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terms

Set Theory Vocabulary

Essential terms and symbols

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concept

Roster Form

Writing sets by listing

Roster Form: Just List Everything

The simplest way to write a set.

List all elements inside curly braces { }.

Separate with commas.

Format: {element1, element2, element3, ...}\{\text{element1, element2, element3, ...}\}

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concept

Set-Builder Form

Writing sets by describing properties

Set-Builder Form: Describe the Rule

Instead of listing, describe what makes something a member.

Format: {x:property of x}\{x : \text{property of } x\}

Or: {xproperty of x}\{x | \text{property of } x\}

The colon (:) or bar (|) means 'such that'

Read: 'The set of all x such that [property] is true'

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practice

Practice: Set Representations

Convert between forms

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Express {2,3,5,7,11,13}\{2, 3, 5, 7, 11, 13\} in set-builder form:

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concept

Empty Set

The set with no elements

Empty Set - The Set With Nothing

A set containing no elements at all.

Symbols: ϕ\phi or {}\{ \}

Number of elements: n(ϕ)=0n(\phi) = 0

Important: ϕ{0}\phi \neq \{0\}

ϕ\phi has zero elements.

{0}\{0\} has one element (the number zero).

Don't confuse them!

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concept

Finite vs Infinite Sets

Understanding set sizes

Finite Sets - Countable

A set where you can count all elements and the counting ends.

Definition: A set with a definite number of elements.

Notation: n(A)n(A) = number of elements in set AA

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quiz

Check: Set Basics

Test your understanding

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The set {x:xN,10<x<11}\{x : x \in \mathbb{N}, 10 < x < 11\} is:

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concept

Equal Sets

When are two sets the same?

When Are Two Sets Equal?

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

Nothing more, nothing less.

Symbol: A=BA = B

Key Point: Order doesn't matter. Repetition doesn't matter.

Only the elements themselves matter.

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concept

Subsets

Understanding containment

What is a Subset?

Set AA is a subset of set BB if every element of AA is also in BB.

Symbol: ABA \subseteq B

Read: "AA is a subset of BB" or "AA is contained in BB"

Think: If BB is a big box and AA is smaller, everything in AA must be in BB.

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concept

Counting Subsets

The 2^n formula

The Subset Formula

If a set has nn elements, it has exactly 2n2^n subsets (including the empty set and the set itself).

Formula: Number of subsets = 2n2^n

This is a powerful counting formula!

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interactive

Subset Visualization

See subsets in action

Subsets of {a, b}abtap to build subset 1All 4 subsets:{ } {a} {b} {a,b}2 elements → 2 × 2 = 4 subsets

Subset Builder

Problem: List all subsets of {a, b}

To build a subset, decide for each element: in or out?

Tap to build all possible subsets.

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quiz

Check: Subsets

Test subset understanding

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If ABA \subseteq B and BCB \subseteq C, then:

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concept

Intervals Introduction

Sets on the number line

Intervals: Special Subsets of Real Numbers

An interval is a set of real numbers between two endpoints.

Think: A section of the number line.

Intervals are subsets of R\mathbb{R} (real numbers).

Why they matter: Used extensively in calculus, inequalities, and real-world measurements.

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concept

Open & Closed Intervals

Understanding boundaries

Open Interval (a,b)(a, b)

Definition: (a,b)={x:a<x<b}(a, b) = \{x : a < x < b\}

Key: Endpoints aa and bb are NOT included.

Use round brackets ()()

Think: Open doors - you can't touch the doors.

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concept

Half-Open & Infinite Intervals

More interval types

Half-Open Intervals

Type 1: [a,b)[a, b)

Left endpoint included, right NOT included.

[a,b)={x:ax<b}[a, b) = \{x : a \leq x < b\}

Type 2: (a,b](a, b]

Left endpoint NOT included, right included.

(a,b]={x:a<xb}(a, b] = \{x : a < x \leq b\}

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concept

Interval Length

Measuring intervals

Length of an Interval

For any finite interval with endpoints aa and bb where a<ba < b:

Length = bab - a

Doesn't matter if endpoints are included or not!

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practice

Practice: Intervals

Master interval notation

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Express {x:x10}\{x : x \geq 10\} in interval notation:

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concept

Universal Set

The context for all sets

Universal Set - The Big Box

The universal set contains all objects under consideration in a particular discussion.

Symbol: UU

Think: The universe for your problem. Everything you're talking about fits inside this box.

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concept

Venn Diagrams

Visualizing set relationships

Venn Diagrams - Pictures of Sets

A Venn diagram uses shapes to represent sets visually.

Invented by: John Venn (1834-1923), British logician

Why they rock: You can SEE relationships between sets instantly!

A B
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interactive

Venn Diagram Explorer

See set relationships visually

ABC123456789A = {1,2,3,4,5} B = {3,4,5,6,7} C = {1,4,5,7,8}tap to place numbers

Three overlapping sets create 7 regions plus the outside.

Given:

  • A = {1, 2, 3, 4, 5}
  • B = {3, 4, 5, 6, 7}
  • C = {1, 4, 5, 7, 8}

Key Operations:

  • A ∩ B ∩ C = {4, 5} — in all three
  • A ∩ B = {3, 4, 5} — shared by A and B
  • A ∩ C = {1, 4, 5} — shared by A and C
  • B ∩ C = {4, 5, 7} — shared by B and C
  • A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

Tap to place each number in its correct region.

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operation

Union of Sets

Combining sets together

Union - Combining Sets

The union of sets AA and BB is everything in AA OR BB (or both).

Symbol: ABA \cup B

Read: "AA union BB"

Think: Merge everything together, no duplicates.

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interactive

Union - Visual

See union in action

UABClick to see A ∪ B

Union Visualization

ABA \cup B includes everything from both sets.

Click to highlight the union!

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operation

Intersection of Sets

Finding common elements

Intersection - Common Elements

The intersection of sets AA and BB contains elements in BOTH AA AND BB.

Symbol: ABA \cap B

Read: "AA intersection BB"

Think: What do they share? The overlap.

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interactive

Intersection - Visual

See intersection highlighted

UABClick to see A ∩ B

Intersection Visualization

ABA \cap B is only the overlapping region.

Click to see it!

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operation

Difference of Sets

Removing elements

Difference - Taking Away

The difference ABA - B contains elements in AA but NOT in BB.

Symbol: ABA - B (also written ABA \setminus B)

Read: "AA minus BB"

Think: Remove from AA anything that's also in BB.

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interactive

Difference - Visual

See difference operation

UABClick: A-B → B-A → Reset

Difference Visualization

See how ABA - B and BAB - A are different!

Order matters in difference.

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quiz

Check: Set Operations

Test union, intersection, difference

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If AB=ϕA \cap B = \phi, then AA and BB are called:

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operation

Complement of a Set

Everything outside the set

Complement - Everything Else

The complement of set AA contains all elements in the universal set UU that are NOT in AA.

Symbol: AA' or AcA^c

Read: "AA complement" or "AA prime"

Think: The opposite. Everything outside AA (but still in UU).

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interactive

Complement - Visual

See complement in action

UAClick to see A' (complement)

Complement Visualization

AA' is everything in UU except AA.

Click to see the complement highlighted!

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theorem

De Morgan's Laws

Complement meets operations

De Morgan's Laws - Powerful Theorems

Named after: Augustus De Morgan (1806-1871)

British mathematician who formalized these laws.

They connect complement with union and intersection.

Used in: Logic, programming, circuit design, probability.

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interactive

De Morgan - Visual Proof

See the law in action

(A ∪ B)'A' ∩ B'ABA'B'Click to see both sides are equal!

De Morgan's Law 1 - Visual Proof

Watch both sides become the same!

Green shows the result on each side.

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reference

Set Properties Reference

All important laws

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assessment

Chapter 1: Final Assessment

Comprehensive chapter test

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Which collection is NOT a set?

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Flashcards

Full Revision: Set Theory Fundamentals

Master set theory basics through 76 flashcards covering definitions, notation, operations, and key laws.

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