If R is a relation defined from set A to set B such that every element in A has at most one image in B and distinct elements in A have distinct images in B, then R is called (a) onto relation (b) one-one relation (c) into relation (d) many-one relation
Answers
1.1.1 Relation
A relation R from a non-empty set A to a non empty set B is a subset of the Cartesian
product A × B. The set of all first elements of the ordered pairs in a relation R from a
set A to a set B is called the domain of the relation R. The set of all second elements in
a relation R from a set A to a set B is called the range of the relation R. The whole set
B is called the codomain of the relation R. Note that range is always a subset of
codomain.
1.1.2 Types of Relations
A relation R in a set A is subset of A × A. Thus empty set φ and A × A are two extreme
relations.
(i) A relation R in a set A is called empty relation, if no element of A is related to any
element of A, i.e., R = φ ⊂ A × A.
(ii) A relation R in a set A is called universal relation, if each element of A is related
to every element of A, i.e., R = A × A.
(iii) A relation R in A is said to be reflexive if aRa for all a∈A, R is symmetric if
aRb ⇒ bRa, ∀ a, b ∈ A and it is said to be transitive if aRb and bRc ⇒ aRc
∀ a, b, c ∈ A. Any relation which is reflexive, symmetric and transitive is called
an equivalence relation.
Note: An important property of an equivalence relation is that it divides the set
into pairwise disjoint subsets called equivalent classes whose collection is called
a partition of the set. Note that the union of all equivalence classes gives
the whole set.
1.1.3 Types of Functions
(i) A function f : X → Y is defined to be one-one (or injective), if the images of
distinct elements of X under f are distinct, i.e.,
x1
, x2 ∈ X, f (x1) = f (x2) ⇒ x1
= x2
. (ii) A function f : X→ Y is said to be onto (or surjective), if every element of Y is the
image of some element of X under f, i.e., for every y ∈ Y there exists an element
x ∈ X such that f (x) = y.
Chapter 1
RELATIONS AND FUNCTIONS
2 MATHEMATICS
(iii) A function f : X→ Y is said to be one-one and onto (or bijective), if f is both oneone and onto.
1.1.4 Composition of Functions
(i) Let f : A → B and g : B → C be two functions. Then, the composition of f and
g, denoted by g o f, is defined as the function g o f : A → C given by
g o f (x) = g (f (x)), ∀ x ∈ A.
(ii) If f : A → B and g : B → C are one-one, then g o f : A → C is also one-one
(iii) If f : A → B and g : B → C are onto, then g o f : A → C is also onto.
However, converse of above stated results (ii) and (iii) need not be true. Moreover,
we have the following results in this direction.
(iv) Let f : A → B and g : B → C be the given functions such that g o f is one-one.
Then f is one-one.
(v) Let f : A→ B and g : B → C be the given functions such that g o f is onto. Then
g is onto.
1.1.5 Invertible Function
(i) A function f : X → Y is defined to be invertible, if there exists a function
g : Y → X such that g o f = I
x
and f o g = IY
. The function g is called the inverse
of f and is denoted by f –1
. (ii) A function f : X → Y is invertible if and only if f is a bijective function.
(iii) If f : X → Y, g : Y → Z and h : Z → S are functions, then
h o (g o f) = (h o g) o f. (iv) Let f : X → Y and g : Y → Z be two invertible functions. Then g o f is also
invertible with (g o f)
–1 = f –1 o g–1
. 1.1.6 Binary Operations
(i) A binary operation * on a set A is a function * : A × A → A. We denote * (a, b)
by a * b. (ii) A binary operation * on the set X is called commutative, if a * b = b * a for every
a, b ∈ X.
(iii) A binary operation * : A × A → A is said to be associative if
(a *
b) *
c = a *
(b *
c), for every a, b, c ∈ A.
(iv) Given a binary operation *
: A × A → A, an element e ∈ A, if it exists, is called
identity for the operation *, if a * e = a = e * a, ∀ a ∈
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