Class 11 Maths • Chapter 01 • Comprehensive Interactive Notes
A set is a well-defined collection of distinct objects. "Well-defined" means that for any object, we can clearly say whether it belongs to the collection or not.
"Collection of all vowels in English alphabet"
(Click to reveal)
We know exactly what they are: {a, e, i, o, u}. This IS a set.
"Collection of best cricket players"
(Click to reveal)
"Best" is subjective. My list might differ from yours. This is NOT a set.
| Form | Description | Example |
|---|---|---|
| Roster (Tabular) | Elements listed within braces { }, separated by commas. | \( A = \{2, 4, 6, 8\} \) |
| Set-Builder | Describes the property possessed by all elements. | \( A = \{x : x \text{ is an even natural number } < 10 \} \) |
The Universal Set (U) is the set that contains all elements under consideration.
The Complement of a set A, denoted by \( A' \), is the set of all elements of U which are not in A.
Example:
If \( U = \{1,2,3,4,5\} \) and \( A = \{1,3\} \), then
\( A' = \{2,4,5\} \)
Two sets A and B are called disjoint if they have no common element.
\( A \cap B = \phi \)
Example:
\( A = \{1,3,5\}, \; B = \{2,4,6\} \)
Set A is a subset of B (\( A \subset B \)) if every element of A is also in B.
Power Set \( P(A) \): The collection of all subsets of A. If \( n(A) = m \), then \( n(P(A)) = 2^m \).
Enter elements separated by commas (max 3 for demo).
Real numbers \( \mathbb{R} \) can be represented as continuous intervals on the real line:
| Interval Type | Notation | Set-Builder Form | Description |
|---|---|---|---|
| Open Interval | \( (a, b) \) | \( \{x : a < x < b\} \) | Excludes endpoints \( a \) and \( b \). |
| Closed Interval | \( [a, b] \) | \( \{x : a \le x \le b\} \) | Includes endpoints \( a \) and \( b \). |
| Semi-Open (L) | \( (a, b] \) | \( \{x : a < x \le b\} \) | Excludes \( a \), includes \( b \). |
| Semi-Open (R) | \( [a, b) \) | \( \{x : a \le x < b\} \) | Includes \( a \), excludes \( b \). |
Length of any interval \( (a,b), [a,b], (a,b] \) or \( [a,b) \) is \( b - a \).
Union (\( \cup \)): Elements in A OR B.
Intersection (\( \cap \)): Elements in BOTH A AND B.
Difference (\( A-B \)): Elements in A but NOT in B.
Symmetric Difference (\( A \Delta B \)): Elements in A or B, but NOT both. Formula: \( A \Delta B = (A-B) \cup (B-A) \).
Enter elements for Set A and Set B.
| Law | Expression |
|---|---|
| Commutative |
\( A \cup B = B \cup A \) \( A \cap B = B \cap A \) |
| Associative | \( A \cup (B \cup C) = (A \cup B) \cup C \) |
| Distributive | \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \) |
| Identity | \( A \cup \phi = A \), \( A \cap U = A \) |
These laws relate union, intersection, and complements.
\( (A \cup B)' = A' \cap B' \)
\( (A \cap B)' = A' \cup B' \)
⚠️ CBSE frequently asks verification using Venn diagrams.
For finite sets A and B:
Let \( A - B \), \( A \cap B \), and \( B - A \) be mutually disjoint sets whose union is \( A \cup B \).
Therefore, \( n(A \cup B) = n(A - B) + n(A \cap B) + n(B - A) \).
Substituting \( n(A - B) = n(A) - n(A \cap B) \) and \( n(B - A) = n(B) - n(A \cap B) \), we get:
\( n(A \cup B) = [n(A) - n(A \cap B)] + n(A \cap B) + [n(B) - n(A \cap B)] \)
\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
When three sets A, B, and C are involved:
✔ Use Venn diagram before substituting values
1. If \( A = \{1, 2\} \), how many subsets does it have?
2. \( A \cup A' \) is equal to:
3. Which set is infinite?
4. If \( A \subset B \), then \( A \cap B \) is:
5. The set of "intelligent students" is: