Relations and Functions

Definitions and Formulas

Relation Definitions:- A relation between two sets is a collection of ordered pairs containing one object from each set. If the object ‘a’ is from set A and the object ‘b’ is from set B, then the objects are said to be related if the ordered pair (a,b) is in the relation. 

i.e. (a,b) ∈ AXB. 

Function Definitions:- A function is a relation from the first set to the second set if each input is related to only one output. i.e. f: A→B

Means, one input is related only to one output but it is possible that two input has the same output.

It is not a function because

(i) c has no output

(ii) a have two outputs but according to the definition each input has only one output.

It is a perfect example of a function because in this relation every element has a unique output.

Types of Relations

(1) Empty Relation:- 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⊂ AXA.

(2) Universal Relation:- 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=AXA.

Note:- Sometimes, both the empty relation and universal relation are called trivial relations.

(3) Reflexive Relation:- A relation R in a set A is called reflexive, if (a, a) ∈ R, for every a ∈ A.

(4) Symmetric Relation:- A relation R in a set A is called symmetric, if (a1, a2) ∈ R, implies that (a2, a1) ∈ R, for all a1, a2 ∈ A.

(5) Transitive Relation:- A relation R in a set A is called transitive, if (a1, a2) ∈ R, and (a2, a3) ∈ R implies that (a1, a3) ∈ R, for all a1, a2, a3 ∈ A.

(6) Equivalence Relation:- A relation R in a set is said to be an equivalence relation if R is reflexive, symmetric, and transitive.

Types of Functions

(1) One-one and Many-one function:- 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, otherwise, f is called many-one. 

This is an example of a one-one function.

For one-one function:- f(x1) = f(x2) implies x1 = x2 for every x1, x2 ∈ X.

This is an example of a many-one function.

(2) Onto Function:- 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 in X such that f(x) = y.

Note:- f:X→Y is onto if and only if Range of f=Y.

(3) Bijective Function:- A function f:X→Y is said to be one-one and onto (or bijective), if f is both one-one and onto.

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