Relations and Functions
Relation
A relation R from a non-empty set A to a non empty set B is a subset of the Cartesianproduct A × B.
The set of all first elements of the ordered pairs in a relation R from aset A to a set B is called the domain of the relation R.
The set of all second elements ina relation R from a set A to a set B is called the range of the relation R.
The whole setB is called the codomain of the relation R.
Note that range is always a subset ofcodomain.
A relation R on set is the subset of A×A.
Therefore,
R ⊂ A × A.
Types of Relations
Extreme Relation
1. Empty
A relation R in a set A is called empty relation, if no element of A is related to anyelement of A, i.e., R = φ ⊂ A × A.
2. Universal
A relation R in a set A is called universal
3. Reflexive
A relation R in A is said to be reflexive if aRa for all a∈A
4. Symmetric
R is symmetric ifaRb ⇒ bRa, ∀ a, b ∈ A
5. Transitive
it is said to be transitive if aRb and bRc ⇒ aRc∀ a, b, c ∈ A
6. Equivalence
Any relation which is reflexive, symmetric and transitive is calledan equivalence relation.
Types of Functions
1. One One or Infective
A function f : X → Y is defined to be one-one (or injective), if the images ofdistinct elements of X under f are distinct, i.e.,x1, x2 ∈ X, f (x1) = f (x2) ⇒ x1 = x2.
2. Onto or Subjective
A function f : X → Y is said to be onto (or surjective), if every element of Y is theimage of some element of X under f, i.e., for every y ∈ Y there exists an elementx ∈ X such that f (x) = y.
3. One One onto or Bijective
A function f : X → Y is said to be one-one and onto (or bijective), if f is both oneone and onto.
Composition of Functions
(i) Let f : A → B and g : B → C be two functions. Then, the composition of f andg, denoted by g o f, is defined as the function g o f : A → C given byg 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. Theng is onto.
1m Invertible Function(i) A function f : X → Y is defined to be invertible, if there exists a functiong : Y → X such that g o f = Ix and f o g = IY. The function g is called the inverseof 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 → h o (g o f) = (h o g) o f.Z and h : Z → S are functions, then(iv) Let f : X → Y and g : Y → Z be two invertible functions. Then g o f is alsoinvertible 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 everya, 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 calledidentity for the operation *, if a * e = a = e * a, ∀ a ∈ A.