INTRODUCTION TO FUNCTION | RELATION & FUNCTION

Function:

A Function is an ordered triple <∱,A,B> such that

1. A and B are sets and ∱ ⊆ A × B
2. For every x ∈A  there is some y ∈ B such that <x,y>∈∱
3.If <x,y>∈∱ and <x,z>∈∱ ,then y=z ;in other words the assignment is unique in the sense that an x ∈A is assigned  atmost one element of B.

• A is called Domain of ∱ and B is called Co-domain.

• Then we will usually write y=∱(x) ,and called " y is the image of x under ∱ "

• The set {y ∈ B :there is an x ∈A such that y=∱(x)}
  is called range of ∱ .

Types Of Function:

One to One Function:

A function f: A → B is One to One if for each element of A there is a distinct element of B. It is also known as Injective. Consider if a₁ ∈ A and b₁ ∈ B, f is defined as f:A→ B such that f (a₁) = f (b₁)

Many to One Function:

It is a function which maps two or more elements of A to the same element of set B. Two or more elements of A have the same image in B.

Onto Function:

If there exists a function for which every element of set B there is (are) preimage(s) in set A, it is Onto Function. Onto is also referred as Surjective Function.

One-One and Onto Function:

A function, f is One–One and Onto or Bijective if the function f is both One to One and Onto function. In other words, the function f associates each element of A with a distinct element of B and every element of B has a pre-image in A.

Special Types Of Function:

Identity Function:

If A = B and f (x) = x for all x ∈ A, the f is called the identity function
on A .

Constant Function:

If f (x) = c for all x ∈ A, then f is called a constant function.

Polynomial Function:

A polynomial function is defined by y =a₀ + a₁x + a₂x² + … + a_nxⁿ, where n is a non-negative integer and a₀, a₁, a₂,…, _n ∈ R.

• Constant Function: If the degree is zero, the polynomial function is a constant function.
• Linear Function: The polynomial function with degree one. Such as y = x + 1
• Quadratic Function: If the degree of the polynomial function is two, then it is a quadratic function
• Cubic Function: A cubic polynomial function is a polynomial of degree three.

Rational Function:

A rational function is any function which can be represented by a rational fraction say, f(x)/g(x) in which numerator, f(x) and denominator, g(x) are polynomial functions of x, where g(x) ≠ 0.

Modulus Function:

The absolute value of any number, c is represented in the form of |c|. If any function f: R→ R is defined by f(x) = |x|, it is known as Modulus Function. For each non-negative value of x, f(x) = x and for each negative value of x, f(x) = -x, i.e.,

f(x) = {x, if x ≥ 0; – x, if x < 0.

Signum Function:

Greatest Integer Function:

If a function f: R→ R is defined by f(x) = [x], x ∈ X. It round-off to the real number to the integer less than the number.It is also called as Floor Function.

For example: [5.12] = 5

Least Integer Function:

If a function f: R→ R is defined by f(x) = ⌈x⌉, x ∈ X. It round-off to the real number to the integer greater than the number. It is also called as Ceiling Function.

For example: [5.12] = 6

Introduction to Relation

Relation:

A relation R from A to B is a subset of the Cartesian product A x B and is derived by describing a relationship between the first element (say x) and the other element (say y) of the ordered pairs in A & B.

Representation of Relation:

A relation is represented either by Roster method or by Set-builder method. Consider an example of two sets A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5}. The relation is that the elements of A are the square of the elements of B.

1. In set-builder form, R = {(x, y): x is the square of y, x ∈ A and y ∈ B}.
2. In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}.

Terminologies (Terms used):

1. Image:
   for any ordered pairs, in any Cartesian product (say A × B), the second element is called the image of the first element.
2. Domain:The set of all first elements of the ordered pairs in a relation R from a set A to a set B.
3. Range:The set of all second elements in a relation R from a set A to a set B is  called Range.
4. Co-domain:
   The whole set B is called Co-domain. Range ⊆ Co-domain.

Total Number of Relations:

For two non-empty set, A and B. If the number of elements in A is h i.e., |A| = h & that of B is k i.e., |B| = k, then the number of ordered pair in the Cartesian product will be |A × B| = hk. The total number of relations is 2^hk.

Types of Relations:

There are many types of relations.

Empty Relation:

If no element of set X is related or mapped to any element of Y, then the relation R in A is an empty relation, i.e, R = Φ.

Universal Relation:

A relation R, called universal relation if each element of A is related to every element of B.

Suppose A is a set of all natural numbers and B is a set of all whole numbers. The relation between A and B is universal as every element of A is in set B.

                    Empty relation and Universal relation are sometimes called trivial relation.

Inverse Relation:

Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R^-1 is said to be an Inverse relation if R^-1 from set B to A is denoted by R^-1 = {(b, a): (a, b) ∈ R}.

Identity Relation:

In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}.

For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.

Reflexive Relation:

If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.

Symmetric Relation:

A relation R on a set A is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation:

A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

Equivalence Relation:

A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.

Ordered Relation:

Partial Ordered Relation:
A relation R⊂X ✖️ X is called partial ordering if it is Reflexive,Anti-Symmetric, and Transitive, that is
   
             (x , x) ∈ R for all x ∈ X
             (x , y) ∈ R ,(y , x) ∈ R ⇒ x = y
             (x , y) ∈ R, (y , z) ∈ R ⇒ (x , z) ∈ R

Total Ordered Relation:
When all the elements of a partial order relation are comparable, the relation is called a total order Relation.
i.e;  a,b ∈ A and  a≤ b and b≤ a
[ ≤ is called relation ]

Linear Ordered:

That is, every element is related with every element one way or the other.
A total order is also called a linear order.

PreOrdered/Quasi Ordered:

A binary relation R on a set A is a quasi order if and only if it is
(1) irreflexive, { (x , x) ∈ R for all x ∉ X }
(2) transitive.  (x , y) ∈ R, (y , z) ∈ R ⇒ (x , z) ∈ R.


(minimal/maximal element):

Let < A, ≤ > be a poset, where ≤ represents an arbitrary partial order. Then an element   b ∈ A is a minimal element of A if there is no element  a ∈ A that satisfies a ≤ b. Similarly an element b ∈ A is a maximal element of A if there is no element a ∈ A that satisfies b ≤ a.
 
(least/greatest element):

Let < A, ≤ > be a poset. Then an element b ∈ A is the least element of A if for every element a ∈ A, b ≤ a.

(well order): A total order R on a set A is a well order if every non-empty subset of A has the least element.