Class 11 Maths Revision Notes for Sets of Chapter 1 â€“ Free PDF Download
Free PDF download of Class 11 Maths revision notes & short key-notes for Sets of Chapter 1 to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books
Chapter Name | Sets |
Chapter | Chapter 1 |
Class | Class 11 |
Subject | Maths Revision Notes |
Board | CBSE |
TEXTBOOK | MatheMatics |
Category | REVISION NOTES |
CBSE Class 11 Maths Revision Notes for Sets of Chapter 1
Set
A set is a well-defined collection of objects.
Representation of Sets
There are two methods of representing a set
- Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by commas.
- Set-builder form In the set-builder form, we list the property or properties satisfied by all the elements of the sets.
Types of Sets â€“ Class 11 Maths Notes
- Empty Sets:Â A set which does not contain any element is called an empty set or the void set or null set and it is denoted by {} or Î¦.
- Singleton Set:Â A set consists of a single element, is called a singleton set.
- Finite and infinite Set:Â A set which consists of a finite number of elements, is called a finite set, otherwise the set is called an infinite set.
- Equal Sets:Â Two sets A and 6 are said to be equal, if every element of A is also an element of B or vice-versa, i.e. two equal sets will have exactly the same element.
- Equivalent Sets:Â Two finite sets A and 6 are said to be equal if the number of elements are equal, i.e. n(A) = n(B)
Subset â€“ Class 11 Maths Notes
A set A is said to be a subset of set B if every element of set A belongs to set B. In symbols, we write
A âŠ† B, if x âˆˆ A â‡’ x âˆˆ B
Note:
- Every set is o subset of itself.
- The empty set is a subset of every set.
- The total number of subsets of a finite set containing n elements is 2^{n}.
Intervals as Subsets of R
Let a and b be two given real numbers such that a < b, then
- an open interval denoted by (a, b) is the set of real numbers {x : a < x < b}.
- a closed interval denoted by [a, b] is the set of real numbers {x : a â‰¤ x â‰¤ b}.
- intervals closed at one end and open at the others are known as semi-open or semi-closed interval and denoted by (a, b] is the set of real numbers {x : a < x â‰¤ b} or [a, b) is the set of real numbers {x : a â‰¤ x < b}.
Power Set
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). If the number of elements in A i.e. n(A) = n, then the number of elements in P(A) = 2^{n}.
Universal Set
A set that contains all sets in a given context is called the universal set.
Venn-Diagrams
Venn diagrams are the diagrams, which represent the relationship between sets. In Venn-diagrams the universal set U is represented by point within a rectangle and its subsets are represented by points in closed curves (usually circles) within the rectangle.
Operations of Sets
Union of sets: The union of two sets A and B, denoted by A âˆª B is the set of all those elements which are either in A or in B or in both A and B. Thus, A âˆª B = {x : x âˆˆ A or x âˆˆ B}.
Intersection of sets:Â The intersection of two sets A and B, denoted by A âˆ© B, is the set of all elements which are common to both A and B.
Thus, A âˆ© B = {x : x âˆˆ A and x âˆˆ B}
Disjoint sets:Â Two sets Aand Bare said to be disjoint, if A âˆ© B = Î¦.
Intersecting or Overlapping sets:Â Two sets A and B are said to be intersecting or overlapping if A âˆ© B â‰ Î¦
Difference of sets:Â For any sets A and B, their difference (A â€“ B) is defined as a set of elements, which belong to A but not to B.
Thus, A â€“ B = {x : x âˆˆ A and x âˆ‰ B}
also, B â€“ A = {x : x âˆˆ B and x âˆ‰ A}
Complement of a set:Â Let U be the universal set and A is a subset of U. Then, the complement of A is the set of all elements of U which are not the element of A.
Thus, Aâ€™ = U â€“ A = {x : x âˆˆ U and x âˆ‰ A}
Some Properties of Complement of Sets
- A âˆª Aâ€™ = âˆª
- A âˆ© Aâ€™ = Î¦
- âˆªâ€™ = Î¦
- Î¦â€™ = âˆª
- (Aâ€™)â€™ = A
Symmetric difference of two sets:Â For any set A and B, their symmetric difference (A â€“ B) âˆª (B â€“ A)
(A â€“ B) âˆª (B â€“ A) defined as set of elements which do not belong to both A and B.
It is denoted by A âˆ† B.
Thus, A âˆ† B = (A â€“ B) âˆª (B â€“ A) = {x : x âˆ‰ A âˆ© B}.
Laws of Algebra of Sets â€“ Class 11 Maths Notes
Idempotent Laws:Â For any set A, we have
- A âˆª A = A
- A âˆ© A = A
Identity Laws:Â For any set A, we have
- A âˆª Î¦ = A
- A âˆ© U = A
Commutative Laws:Â For any two sets A and B, we have
- A âˆª B = B âˆª A
- A âˆ© B = B âˆ© A
Associative Laws:Â For any three sets A, B and C, we have
- A âˆª (B âˆª C) = (A âˆª B) âˆª C
- A âˆ© (B âˆ© C) = (A âˆ© B) âˆ© C
Distributive Laws:Â If A, B and Care three sets, then
- A âˆª (B âˆ© C) = (A âˆª B) âˆ© (A âˆª C)
- A âˆ© (B âˆª C) = (A âˆ© B) âˆª (A âˆ© C)
De-Morganâ€™s Laws:Â If A and B are two sets, then
- (A âˆª B)â€™ = Aâ€™ âˆ© Bâ€™
- (A âˆ© B)â€™ = Aâ€™ âˆª Bâ€™
Formulae to Solve Practical Problems on Union and Intersection of Two Sets
Let A, B and C be any three finite sets, then
- n(A âˆª B) = n(A) + n (B) â€“ n(A âˆ© B)
- If (A âˆ© B) = Î¦, then n (A âˆª B) = n(A) + n(B)
- n(A â€“ B) = n(A) â€“ n(A âˆ© B)
- n(A âˆª B âˆª C) = n(A) + n(B) + n(C) â€“ n(A âˆ© B) â€“ n(B âˆ© C) â€“ n(A âˆ© C) + n(A âˆ© B âˆ© C)