Types of Relations1. Reflexive Relation – A relation R in set A is called reflexive if (a,a)Є R for every a Є A
2. Symmetric Relation-A relation r in a set A is called symmetric if (a,b) ЄR implies that (b,a) Є R
For every a,b Є R
3.Transitive relation-If (a,b ) Є R and (b,c) Є R implies that (a,c) Є R for every a,b,c ЄA
Equivalence relation- A relation R in a set A is said to be equivalence relation if R is reflexive,
Symmetric and transitive. An equivalence relation defined on set A divides it into disjoint subsets
These disjoint subsets are called equivalence class.
Functions1.one-one or injective functions- a function f:X→Y is defined to be one –oneif the images of
distinct element of X under f are distinct.
2. onto or surjective function- a function f : X → Y is said to be onto if every element of Y is the
image of some element of X.
Bijective function- A function f : X → Y is said to bijective if it is one-one and onto both.
Composition of functions and invertible functions- Let f : A → B and g : B f : X → Y C,then the
composition of f and , dnoted by gof, is defind as the function gof: A → C given by
gof(x) =g(f(x)) for every x Є A
Remarks-1.The composition of two one-one functions is one-one .
2. The composition of two onto functions is onto .
Definition- A function f: X → Y is dfined to be invertible, if there exists afunction g : Y → X such
that gof = Ix and fog =Iy . The function g is called the inverse of f and is denoted by f-1.
Thus if f is invertible, then f must be one-one and onto and conversely, if f is one-one and
onto,then f must be invertible .
Binary operation-A binary operation * on a set A is afunction * : A X A → A. We denote *(a,b) by
a*b.
A binary operation * on set A is said to commutative if a * b = b*a for every a,bЄ A
A binary operation * on set A is said to be associative if (a*b)*c = a*(b*c) for every a, b,cЄA
Identity element-for a given opration * :A X A→A,an element e Є A,if it exists,is called identity for
operation *,if a*e = e *a for every a ЄA
Inverse element-.For agiven binary operation * onset A with identity element e in set A ,an element a is said to
be invertible if there exists an element b such that a*b = b*a =e and bis called inverse of b