Relations and functions notes

1. CHAPTER – 1
RELATIONS& FUNCTIONS
Basic concepts andformulae:
 Relation : If A and B are two non- empty sets, then any subset R of A ×B is called
relation from set A to set B. i.e. R : A → 𝐵 ⇔ 𝑅 ⊆ A × B.
 Some standard types of Relations : Let A be a non- empty set. Then, a relation R on set A
is said to be.
 Reflexive : If (x, x)∈ R for each element x∈ A, i.e., if x R x for each element x ∈ A.
 Symmetric : If (x, y)∈ R ⇒ (y, x) ∈ R for all x, y∈ A, i.e., if x Ry ⇒ y Rx for all x, y ∈ A.
 Transitive : If (x, y)∈ R and (y, z)∈ R ⇒ (x, z) ∈, for all x, y, z ∈ A, i.e., if xRy and yRz
⇒ xRz.
 Equivalence relation : Any relation R on a set A is said to be an equivalence relation if
R is reflexive, symmetric and transitive.
 A function F:X→Yis one -one or injective if f(x1)=f(x2) ⟹x1= x2 for all x1, x2∈
X
 A function F:X→Yis onto or surjectiveif for every y∈ Y there exist x∈ X:
f(x)=y
 A function which is both one-one and onto is called bijective.
 A function F:X→Yis invertible if and only if f is bijective.
 The composition of functions f: A→B and g: B→C is the functiopn gof: A→C
given by gof(x)= g(f(x))
 A function F:X→Yis invertible if there exist g: Y→X such thast gof=Ix and
fog= Iy.
 A binary operation * ona set A is a function fromA×A to A.
 An operation * on X is commutative if a*b=b*a for all a,b∈ X
 An operation * on X is Associativeif (a*b)*c=a*(b*c) for all a,b,c ∈ X
 An element e∈ X is the identity element for the binary operatuion * if
a*e=a=e*a for all a∈ X
 An element a∈ X is invertible for binary operation* if there existb∈
Xsuch that a*b=b*a=e , where e is the identity for the binary operation*.
The element bis calledinverse of a and is denotedby a-1
.