how many types of relation and function
Answers
Answer:
in maths their are nine kindes of relation
Step-by-step explanation:
1) empty relation .
2)full relation .
3)reflexive relation .
4)iireflexive relation .
5)transitive relation.
6)symmetric relation.
7)antysymmetric relation .
8)asymmetric relation.
9)equivalence relation.
Step-by-step explanation:
➡️RELATION
==> A relation R from setA to setB in a subset of a set A×B
e.g. => A = { 1, 2, 3 } and B = {a, b}
A×B = { (1,a) , (1,b) , (1,c) , (2,a) , (2,b) , (2,c) }
R = { (1,a) , (2,a) , (2,b) , (2,c) } ----> It is a realtion
R1 = { (2,a) , (2,b) , (b,1) } ------> It is not a relation due to (b,1) because realtion is subset of A×B .
TYPES OF RELATION ON SET A
(1) IDENTITY RELATION
==> A realtion R on setA is said to be an identity relation if for all x belonging to A , x has R relation on with x ( xRx ).
eg. A = A = { 1, 2, 3 }
A×A = A = { (1,1) , (1,2) , (1,3) ,(2,1) , (2,2) , (2,3) , (3,1) ,(3,2) , (3,3) }
R = { (1,1) , (2,2) , (3,3) } -- it is an identity relation
(2) REFLEXIVE RELATION
==> A relation R on setA said to be reflexive relation , if xRx or (x,x) belonging to R for all x belonging to A.
eg. A = A = { 1, 2, 3 }
A×A = A = { (1,1) , (1,2) , (1,3) ,(2,1) , (2,2) , (2,3) , (3,1) ,(3,2) , (3,3) }
R = { (1,1) , (2,2) , (1,3) , (3,3) } ---- it is a reflexive relation.
(3) SYMMETRIC RELATION
==> A realtion R on setA is said to be symmetric, if xRx => yRx , where x,y belonging to A.
eg. A = A = { 1, 2, 3 }
A×A = A = { (1,1) , (1,2) , (1,3) ,(2,1) , (2,2) ,
(2,3) , (3,1) ,(3,2) , (3,3) }
R = (1,1) , (2,3) , (3,2) , (2,2) , (3,3) } --- it is a symmetric relation.
(4) ANTISYMMETRIC RELATION
==> A realtion R on setA is said to be antisymmetric if xRy and yRx => "x = y" .where x,y belonging to A.
eg. A = A = { 1, 2, 3 }
A×A = A = { (1,1) , (1,2) , (1,3) ,(2,1) , (2,2) , (2,3) , (3,1) ,(3,2) , (3,3) }
R = { (1,1) , (2,2) , (3,3) } ---> it is an an antisymmetric relation.
(5) TRANSITIVE RELATION
==> A realtion R on setA is said to be transitive if xRy and yRz => xRz . where x,y,z belonging to A.
eg. A = A = { 1, 2, 3 }
A×A = A = { (1,1) , (1,2) , (1,3) ,(2,1) , (2,2) , (2,3) , (3,1) ,(3,2) , (3,3) }
R = { (1,1) , (2,2) } -- it is a transitive relation.
(6) EQUIVALENCE RELATION
==> A realtion R on setA is said to be an equivalence relation if it is reflexive , symmetric as well as transitive.
eg. A = A = { 1, 2, 3 }
A×A = A = { (1,1) , (1,2) , (1,3) ,(2,1) , (2,2) , (2,3) , (3,1) ,(3,2) , (3,3) }
R = { (1,1) , (2,3) , (3,2) , (1,3) ,(2,2) , (3,3) }
it is reflexive, symmetric & transitive so this relation is equivalence relation.
NOTE :
EVERY IDENTITY RELATION IS REFLEXIVE BUT EVERY REFLEXIVE RELATION IS NOT TO BE AN IDENTITY RELATION.
➡️ FUNCTION
==> A function 'F' from setA to setB is a relation (or association) that associates each element of setA with a unique element of setB.
usually we write F : A ---> B
Here ,
setA is known as domain of function 'F'
and setB is known as codomain of function 'F'.
➡️ TYPES OF FUNCTION :
(1) ONTO ( or , SUBJECTIVE ) FUNCTION
==> A function whose range is equal to its codomain is known as 'onto' function.
(2) INTO FUNCTION
==> If range is not equal to ita Coco then the function is known as 'into' function.
(3) ONE - ONE (or, INJECTIVE ) FUNCTION
==> If each element of setA is associated with a separate element of setB then it is known as 'one-one' function.
Otherwise known as 'many - one' function.
(4) BIJECTIVE FUNCTION
==> A function which is both one-one (injective) and onto (surjective) is known as bijective function.