Generated with Monsha

Save this resource to edit, expand, or export it, or create more resources for free.

Class 11 Maths: Chapter 1 - Sets (NCERT Karnataka Board)

Anything
PU -2
English

Class 11 Maths: Chapter 1 - Sets (NCERT Karnataka Board)

1. Introduction and Definition (ಪೀಠಿಕೆ ಮತ್ತು ವ್ಯಾಖ್ಯೆ)

A Set is a well-defined collection of objects. The objects, elements, and members of a set are synonymous terms.
ಗಣ (Set): ಸುಸ್ಪಷ್ಟವಾಗಿ ನಿರ್ದರಿಸಬಹುದಾದ ವಸ್ತುಗಳ ಸಮೂಹವನ್ನು 'ಗಣ' ಎನ್ನುತ್ತೇವೆ.

Key Points:

  • Sets are usually denoted by capital letters:
    
    .
  • Elements are represented by small letters:
    
    .
  • If
    
    is an element of set
    
    , we write
    
    (read as "a belongs to A").
  • If
    
    is not an element of set
    
    , we write
    
    (read as "b does not belong to A").


2. Representation of Sets (ಗಣಗಳ ನಿರೂಪಣೆ)

There are two methods of representing a set:
  1. Roster or Tabular Form (ಪಟ್ಟಿ ವಿಧಾನ): All elements are listed, separated by commas, and enclosed within braces
    
    .
  • Example: The set of vowels in English:
    
    .
  1. Set-builder Form (ಗಣ ನಿರ್ಮಾಣ ರೂಪ): All elements possess a single common property which is not possessed by any element outside the set.
  • Example:
    
    .


3. Types of Sets (ಗಣಗಳ ವಿಧಗಳು)

Type
Definition
Kannada Term
Empty Set
A set which does not contain any element. Denoted by

or

.
ಶೂನ್ಯ ಗಣ
Finite Set
A set which is empty or consists of a definite number of elements.
ಪರಿಮಿತ ಗಣ
Infinite Set
A set which is not finite (contains unlimited elements).
ಅಪರಿಮಿತ ಗಣ
Equal Sets
Two sets

and

having exactly the same elements.
ಸಮ ಗಣಗಳು
Singleton Set
A set containing only one element.
ಏಕಾಂಶ ಗಣ


4. Subsets and Power Set (ಉಪಗಣಗಳು ಮತ್ತು ಘಾತ ಗಣ)

Subsets (ಉಪಗಣ)

Set

is said to be a subset of set

if every element of

is also an element of

.
  • Symbol:
    
    .
  • Every set is a subset of itself:
    
    .
  • Empty set is a subset of every set:
    
    .

Power Set (ಘಾತ ಗಣ)

The collection of all subsets of a set

is called the Power Set of

, denoted by

.
  • If a set
    
    has
    
    elements, then the number of elements in its power set is
    
    .
  • 
    .

Universal Set (ವಿಶ್ವ ಗಣ)

A set that contains all objects, including itself, is called the Universal Set, denoted by

.


5. Operations on Sets (ಗಣಗಳ ಮೇಲಿನ ಕ್ರಿಯೆಗಳು)

Union of Sets (ಗಣಗಳ ಸಂಯೋಗ)

The union of two sets

and

is the set which consists of all the elements of

and all the elements of

.
  • Symbol:
    
  • 
fig 1: Venn diagram showing the Union of sets A and B (A ∪ B) with both circles shaded.
fig 1: Venn diagram showing the Union of sets A and B (A ∪ B) with both circles shaded.


Intersection of Sets (ಗಣಗಳ ಛೇದನ)

The intersection of sets

and

is the set of all elements which are common to both

and

.
  • Symbol:
    
  • 
fig 2: Venn diagram showing the Intersection of sets A and B (A ∩ B) with the overlapping region shaded.
fig 2: Venn diagram showing the Intersection of sets A and B (A ∩ B) with the overlapping region shaded.


Difference of Sets (ಗಣಗಳ ವ್ಯತ್ಯಾಸ)

The difference of the sets

and

is the set of elements which belong to

but not to

.
  • Symbol:
    
  • 
fig 3: Venn diagram showing the Difference of sets A - B with the region unique to A shaded.
fig 3: Venn diagram showing the Difference of sets A - B with the region unique to A shaded.


Complement of a Set (ಪೂರಕ ಗಣ)

Let

be the universal set and

be a subset of

. The complement of

is the set of all elements of

which are not the elements of

.
  • Symbol:
    
    or
    
  • 
fig 4: Venn diagram showing the Complement of set A (A') with the region outside circle A shaded.
fig 4: Venn diagram showing the Complement of set A (A') with the region outside circle A shaded.



6. Important Formulae (ಪ್ರಮುಖ ಸೂತ್ರಗಳು)

For any two finite sets

and

:
  1. Cardinal Number Formula:
    
  1. If
    
    and
    
    are disjoint sets (i.e.,
    
    ):
    
  1. For three sets
    
    and
    
    :
    
  1. De Morgan's Laws:
  • 
  • ( (A \cap B)' = A' \cup B' \