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Part of NCERT 11 Mathematics
Make sure you're ready to start
See sets in your world
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.
What makes a set a set?
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.
Essential terms and symbols
Writing sets by listing
The simplest way to write a set.
List all elements inside curly braces { }.
Separate with commas.
Format:
Writing sets by describing properties
Instead of listing, describe what makes something a member.
Format:
Or:
The colon (:) or bar (|) means 'such that'
Read: 'The set of all x such that [property] is true'
Convert between forms
Express in set-builder form:
The set with no elements
A set containing no elements at all.
Symbols: or
Number of elements:
Important:
has zero elements.
has one element (the number zero).
Don't confuse them!
Understanding set sizes
A set where you can count all elements and the counting ends.
Definition: A set with a definite number of elements.
Notation: = number of elements in set
Test your understanding
The set is:
When are two sets the same?
Two sets are equal if they have exactly the same elements.
Nothing more, nothing less.
Symbol:
Key Point: Order doesn't matter. Repetition doesn't matter.
Only the elements themselves matter.
Understanding containment
Set is a subset of set if every element of is also in .
Symbol:
Read: " is a subset of " or " is contained in "
Think: If is a big box and is smaller, everything in must be in .
The 2^n formula
If a set has elements, it has exactly subsets (including the empty set and the set itself).
Formula: Number of subsets =
This is a powerful counting formula!
See subsets in action
Problem: List all subsets of {a, b}
To build a subset, decide for each element: in or out?
Tap to build all possible subsets.
Test subset understanding
If and , then:
Sets on the number line
An interval is a set of real numbers between two endpoints.
Think: A section of the number line.
Intervals are subsets of (real numbers).
Why they matter: Used extensively in calculus, inequalities, and real-world measurements.
Understanding boundaries
Definition:
Key: Endpoints and are NOT included.
Use round brackets
Think: Open doors - you can't touch the doors.
More interval types
Type 1:
Left endpoint included, right NOT included.
Type 2:
Left endpoint NOT included, right included.
Measuring intervals
For any finite interval with endpoints and where :
Length =
Doesn't matter if endpoints are included or not!
Master interval notation
Express in interval notation:
The context for all sets
The universal set contains all objects under consideration in a particular discussion.
Symbol:
Think: The universe for your problem. Everything you're talking about fits inside this box.
Visualizing set relationships
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!
See set relationships visually
Three overlapping sets create 7 regions plus the outside.
Given:
Key Operations:
Tap to place each number in its correct region.
Combining sets together
The union of sets and is everything in OR (or both).
Symbol:
Read: " union "
Think: Merge everything together, no duplicates.
See union in action
includes everything from both sets.
Click to highlight the union!
Finding common elements
The intersection of sets and contains elements in BOTH AND .
Symbol:
Read: " intersection "
Think: What do they share? The overlap.
See intersection highlighted
is only the overlapping region.
Click to see it!
Removing elements
The difference contains elements in but NOT in .
Symbol: (also written )
Read: " minus "
Think: Remove from anything that's also in .
See difference operation
See how and are different!
Order matters in difference.
Test union, intersection, difference
If , then and are called:
Everything outside the set
The complement of set contains all elements in the universal set that are NOT in .
Symbol: or
Read: " complement" or " prime"
Think: The opposite. Everything outside (but still in ).
See complement in action
is everything in except .
Click to see the complement highlighted!
Complement meets operations
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.
See the law in action
Watch both sides become the same!
Green shows the result on each side.
All important laws
Comprehensive chapter test
Which collection is NOT a set?
Master set theory basics through 76 flashcards covering definitions, notation, operations, and key laws.