TYPES OF RELATIONS

EMPTY RELATIONS - An empty relation is a type of relation, there is no relation between any object / element of sets. Empty relation is also called void relation.
Therefore, R =  ɸ 
EXAMPLE - A = Set of all student of boys school
R = {(a, b) ; a and b are sisters}.
UNIVERSAL - A relation R in a set A is called universal relation if each element of set A is related to every element of A. Universal Relation is also called full relation.
Therefore, R = A❌A.
EXAMPLE -  R = {(a, b) : a and b is greater than 2 feet}.
IDENTITY - A relation R in a set A is called identity relation if every element of set A is related to itself only.
Therefore, R = I {(a,a) belongs to A}.
INVERSE - Let R be a relation from set A to set B.
Therefore R belongs to A❌B.
The relation R-1 is called inverse relation, if relation from set B to A is denoted by R-1 = {(b, a) : a and b belongs to R}.
EXAMPLE - R = {(1, 2), (2,3)}
R-1 = {(2,1), (3,2)}.
REFLEXIVE - If every element of set A maps to itself.
for every a ∈ A, (a, a) ∈ R.
SYMMETRIC - A relation R in a set A is said to be symmetric if (a,b) ∈ R then (b,a) ∈ R.
TRANSITIVE - A relation in a set A is said to be transitive, if 
(a,b) ∈ R , (b, c) ∈ R , then (a, c) ∈ R.
EQUIVALENCE - A relation is said to be equivalence relation if it is reflexive, symmetric and transitive relation.
EXAMPLE - If we throw two dices A and B and note down all the possible outcome.
Define, a relation; R = {(a, b) : a ∈ R , b ∈ R}
we find that {(1,1), (2, 2), (3, 3), (4, 4),......} ∈ R (Reflexive)
If {(a,b) = (1,2) ∈ R} then {(B,A) = (2,1)} ( Symmetric)
IF {(a,b) = (1, 2) ∈ R }, {(b,c) = (2,3) ∈ R } 
then  {(a,c ) = (1,3)  ∈ R } (Transitive ).

Definition of relation with example.

RELATIONS - A relation between two sets in a collection of ordered pair containing one object from each set, if the object x take from first set and object y take from second set then obtained ordered pair (x, y) is in relation. The sets must not be empty.
In other words, A connection between two or more sets is a relation. 
NOTE - Functions are type of relation.

EXAMPLE - Given example we can see objects of  first set is related to objects of second set.
Here, 
R = {( -1, 1), (1, 1), (7, 49), (.5, .49)}.

MAPPING,FUNCTION, DOMAIN, CO-DOMAIN AND RANGE DEFINITION WITH EXAMPLES

 MAPPING - Each element of a given set ( the domain of the function) is associated with an element of another set (the range of the function) is called mapping. Some mappings are represent in the above structure.

Function - A function is a relation for which each value from the set the first components of the ordered pair is associated with exactly one value from the set of second components of the ordered pair. 

DOMAIN - The domain of a function f(x) is the set of all values for which the function is defined.

RANGE - The range of the functions is the set of all values that f takes.

Above we have two sets A and B where
 A = (-1, 1, 7, -5 ) ;
                                                                 
 B =  (1, 49, 50 )                                                                                                                             A                    B
here  function  f : A→ B ; A is domain and
                                   B is co-domain             
we know that Range is subset of Co-Domain.                                                                                                                   
So,
f (-1) = f (1) = 1
f (7) = 49
f (50 = 25.

In this example range = co-domain.
Range = (1, 49, 25 )

•  In given example we can see X is domain and Y is a co-domain.

We can write, Domain = (1, 2, 3 )
                     Co-domain = (1, 2, 3, 4 )
                             Range = (1, 2, 3 ).

CLASS 12 COURSE STRUCTURE

      SYLLABUS 2020 - 2021

           CLASS - 12

        COURSE STRUCTURE

     NUMBER                 UNITS                                                                                 Marks

1. Relation and function                                                                       08
2. Algebra                                                                                            10
3. Calculus                                                                                           35
4. Vector and Three - Dimensional Geometry                                     14
5.  Linear Programming                                                                       05
6.  Probability                                                                                       08

            Total                                                                                                80 

                                                                                                 

       Internal Assessment                                                                           20

RELATIONS AND FUNCTIONS

 RELATION - A RELATION between two sets is a collection of ordered pairs containing one object from each sets.

If the object x from the set A and the object y from set B, then the objects are said to be related if the ordered pair (x,y) is in the relation. A function is a type of relation.

Some Examples

TWIN PRIME, PRIME TRIPLETS, CO- PRIME NUMBERS

                                            MORE ABOUT NUMBERS:

1) TWIN PRIME - Prime numbers differing by two are called twin prime number.
example - 5 and 7; 11 and 13 etc.
Solution- 7-5=2 and 13-11=2.

2) PRIME TRIPLETS - The set of three consecutive prime numbers with a difference two is called the prime triplets.

Note: (1, 3, 5) is only prime triplet set in prime numbers.

3) CO-PRIME - The pairs of numbers, which is only divisible by one are called co- prime numbers.
example - 10 and 21 are co- prime numbers ; 16 and 25 are co- prime numbers etc.

PRIME NUMBERS AND COMPOSITE NUMBERS

               

PRIME NUMBER - A natural number, which is divisible by 1 and itself and always greater than 1 is called prime number.

example-2, 3, 5, 7, 11, 13......and so on.

here, 2 is only even prime number, we can say Except 2, every prime number is an odd natural number.

1) Smallest even prime number is 2.
2) Smallest odd prime number is 3.

Largest prime number we can not obtained because prime number are infinitely exist.

COMPOSITE NUMBER - A natural number, which is greater than 1 and not prime.In other words we can say natural number which is divisible by more than two number, is called composite number.

example - 4, 6, 8, 9, 10,.....etc.

composite number can be even or odd.

1)Smallest composite number is 4.

2)Largest composite number we can not obtained because composite number are infinitely exist.

note - 1 is neither prime nor composite.