notes of relations and functions

relation and function

1. What is Cartesian Sets?

Given two non-empty sets A and B. The Cartesian product A×B is the set of all ordered pairs of elements from A and B, i.e.,
A×B={(a,b):a∈A,b∈B}
If either A or B is the null set, then A×B will also be empty set, i.e., A×B

Important tips

1. Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.
2. If there are p elements in A and q elements in B, then there will be pq elements in A×B, i.e., if n(A)=p and n(B)=q, then n(A×B)=pq.
3. If A and B are non-empty sets and either A or B is an infinite set, then so is A×B.
4. A×A×A={(a,b,c):a,b,c∈A}. Here (a,b,c) is called an ordered triplet

Questtion 1 If P={1,2} and Q={5,4,2}, find P×Q and Q×P.
Solution
P={1,2} and Q={5,4,2}
We know that the Cartesian product P×Q of two non-empty sets P and Q is defined as
P×Q={(p,q):p∈P,q∈Q}
∴P×Q={(1,5),(1,4),(1,2),(2,5),(2,4),(2,2)}
Q×P={(5,1),(5,2),(4,1),(4,2),(2,1),(2,2)}

Question 2 If A={−1,1}, find A×A×A
Solution
It is known that for any non-empty set A, A×A×A is defined as
A×A×A={(a,b,c):a,b,c∈A}
It is given that A={−1,1}
∴A×A×A{(−1,−1,−1),(−1,−1,1),(−1,1,−1),(−1,1,1),(1,−1,−1),(1,−1,1),(1,1,−1),(1,1,1),}

2. What is relations?

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A×B.
It "maps" elements of one set to another set. The subset is derived by describing a relationship between the first element and the second element of the ordered pair (A×B).
Domain: 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.
Range: the set of all the ending points is called the range
A relation can be expressed in Set builder or Roaster form

Roster forms

In a Roster forms, all the elements in the set is listed.
Example
Set of vovel={a,e,i,o,u}

Some Important points

• In roster form, the order in which the elements are listed is immaterial
• while writing the set in roster form an element is not generally repeated

Set Builder Form

• In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set {2,4,6,8}, all the elements possess a common property, namely, each of them is a even number less than 10. Denoting this set by N, we write
  N = {x : x is a even number less than 10 }
• b) We describe the element of the set by using a symbol x (any other symbol like the letters y, z, etc. could be used) which is followed by a colon “ : ”. After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces

Important Note

The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A⋅B. If n(A)=p and n(B)=q, then n(A⋅B)=pq and the total number of relations is 2pq

Example:
Let P={1,2,3,.....,18} define a relation R from P to P by R={(x,y):2x−y=0,wherex,y∈P} Write down its domain, codomain and range.
Solution: The relation R from P to P is given as
R = {(x,y):2x-y=0, where x, y ∈ P}
i.e., R = {(x, y): 2x = y, where x, y ∈ P}

∴R={(1,2),(2,4),(3,6),(4,8),(5,10),(6,12),(7,14),(8,16),(9,18)}
The domain of R is the set of all first elements of the ordered pairs in the relation.
∴DomainofR={1,2,3,4,5,6,7,8,9}
The whole set P is the codomain of the relation R.
Therefore codomain of R=P={1,2,3,…,18}
The range of R is the set of all second elements of the ordered pairs in the relation.
Therefore range of R={2,4,6,8,10,12,14,16,18}

3. What is Function

• A function is a "well-behaved" relation
• A function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in fhave the same first element.
• For a relation to be a function, there must be only and exactly one y that corresponds to a given x
• If f is a function from A to B and (a,b)∈f, then f(a)=b, where b is called the image of a under f and a is called the preimage of bunderf.

Example 1:
Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

1. {(3,1),(5,1),(7,1),(11,1),(14,1),(17,1)}
2. {(2,1),(4,2),(6,3),(6,4),(10,5),(12,6),(14,7)}
3. {(1,3),(1,5),(2,5)}

Answer

1. {(3,1),(5,1),(7,1),(11,1),(14,1),(17,1)}
   Since 3, 5, 8, 11, 14, and 17 are the elements of the domain of the given relation having their unique images, this relation is a function.
2. {(2,1),(4,2),(6,3),(6,4),(10,5),(12,6),(14,7)}
   Since the same first element i.e 6 corresponds to two different images 3 and 4, this relation is not a function
3. {(1,3),(1,5),(2,5)}
   Since the same first element i.e., 1 corresponds to two different images i.e., 3 and 5, this relation is not a function.

Important functions

Let us take some useful polynomial and shapes obtained on the Cartesian plane

S.No.	y=p(x)	Graph obtained	Name of the graph	Name of the function
1.	y=mx+c where m and c can be any values (m≠0) Example y=2x+3		Graphs of these functions are straight lines. m is the slope and b is the yintercept. If m is positive then the line rises to the right and if m is negative then the line falls to the right	Linear function. Typical use for linear functions is converting from one quantity or set of units to another.
2.	y=ax2+bx+c where, b2−4ac>0 , a≠0 and a>0 example- y=x2−7x+12		Parabola It intersect the x- axis at two points Example- (3,0) and (4,0)	Quadratic function
3.	y=ax3+bx2+cx+d where,a≠0	It can be of any shape	It will cut the x-axis at the most 3 times	Cubic Function
4.	anxn+an−1xn−1+an−2xn−2+…………+ax+a0 where an≠0	It can be of any shape	It will cut the x-axis at the most n times	Polynomial function
5.	y=f(x)g(x) where g(x)≠0 example- y=1x	It can be any shape	An asymptote is a line that the curve approaches but does not cross.There are vertical and horizontal asymptote	Rational function
6.	y=|x| i.e., y=−x for x<0 y=x for x≥0			Modulus function
7.	y=aln(x)+b where x is in the natural logarithm and a and b are constants They are only defined for positive		For small x they are negative and for large x they are positive x	Logarithmic functions
8.	y=[x]  [x]−>the value of the greatest integer, less than or equal to x			Greatest integer function

4. Algebra of Real Function

Real Value Function: A function which has all real number or subset of the real number as it domain
Real Valued Function: A function which has all real number or subset of the real number as it range
For functions f:X−>R and g:X−>R, we have

1. Addition
   (f+g)(x)=f(x)+g(x),x∈X
2. Substraction
   (f−g)(x)=f(x)−g(x),x∈X
3. Multiplication
   (f.g)(x)=f(x).g(x),x∈X
4. Multiplication by real number (kf)(x)=kf(x),x∈X, where k is a real number. 
5. Division
   fg(x)=f(x)g(x)
   x∈X and g

notes for trigonometry

                                    trignometry


1. Introduction

Trigonometry (from Greek trigõnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3^rd century BC from applications of geometry to astronomical studies.
Trigonometry is most simply associated with planar right angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles

What is angle

An angle which has its vertex at the origin and one side lying on the positive x-axis. It can have a measure which positive or negative and can be greater than 360°

1. If the direction of rotation is anticlockwise, angle is positive . If the direction of rotation is clockwise,angle is negative
2. Once you have made a full circle (360°) keep going and you will see that the angle is greater than 360° .In fact you can go around as many times as you like. The same thing happens when you go clockwise. The negative angle just keeps on increasing
3. It can be measured in degrees or radian

Degree and Radian

They both are unit of measurement of angles
Radian: A unit of measure for angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.
Degree: If a rotation from the initial side to terminal side is (1/360) of a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is called a minute, written as 1", and one sixtieth of a minute is called a second, written as 1'.
Thus, 1° = 60', 1' = 60"
Relation between Degree and Radian
2π radian = 360 ° π radian= 180 ° 1 radian= (180/π) °

Degree	30°	45°	60°	90°	120°	180°	360°
Radian	π/6	π/4	π/3	π/2	2π/3	π	2π

Trigonmetric Ratio's

In a right angle triangle ABC where B=90° ,we can define six ratio's for the two sides i.e Hypotenuse/Base, Base/Perpendicular,Perpendicular/Base,Base/hypotenuse,Hypotenuse/Perpendicular,Perpedicular/Hyptenuse Trignometric ratio's are defined as
sin θ= Perpendicular/Hypotenuse 
cosec θ= Hypotenuse/Perpendicular
cos θ= Base/Hypotenuse 
sec θ= Hypotenuse/Base
tan θ= Perpendicular/Base 
cot θ= Base/Perpendicular
Notice that each ratio in the right-hand column is the inverse, or the reciprocal, of the ratio in the left-hand column.
The reciprocal of sin θ is csc θ ; and vice-versa. 
The reciprocal of cos θ is sec θ.
And the reciprocal of tan θ is cot θ
These are valid for acute angles.
We are now going to define them for any angles and they are called now the Trigometric functions.

Trignometric functions:

Consider a unit circle with center at the origin O and Let P be any point on the circle with P(a,b). And let call the angle x We use the coordinates of P to define the cosine of the angle and the sine of the angle. Specifically, the x-coordinate of B is the cosine of the angle, and the y-coordinate of B is the sine of the angle. Also it is clear
a²+b²=1
cos²x +sin²x =1

Properties of these functions

1. Sine and cosine are periodic functions of period $360^{\circ}$, that is, of period $2\pi $. That's because sines and cosines are defined in terms of angles, and you can add multiples of $360^{\circ}$, or $2\pi $, and it doesn't change the angle. Thus, for any angle x
   sin(x+2π)= sin (x) and cos(x+2π)= cos(x)
   or we can say that
   sin (2nπ + x) = sin x, $n\in Z$ , cos (2nπ + x) = cos x, $n\in Z$ 
   Where Z is the set of all integers
2. sin x = 0 implies x = nπ, where n is any integer
   cos x = 0 implies x = (2n + 1)(π/2)
3. The other trignometric function are defined as
   cosec(x)= 1/sin (x) where x ≠nπ, where n is any integer
   sec(x)=1/cos (x) where x ≠(2n + 1)(π/2) where n is any integer
   tan(x)=sin(x)/cos(x) where x ≠(2n + 1)(π/2) where n is any integer
   cot(x)=cos(x)/sin(x) where x ≠nπ, where n is any integer
4. For all real x
   sin²(x)+cos²(x)=1
   1+ tan²(x)=sec²(x)
   1+ cot²(x)=cosec²(x)
5. What is is Odd function and Even Function
   We have come across these adjectives 'odd' and 'even' when applied to functions, but it's important to know them. A function f is said to be an odd function
   if for any number x, f(-x) = -f(x).
   A function f is said to be an even function if for any number x, f(-x) = f(x).
   Many functions are neither odd nor even functions, but some of the most important functions are one or the other.
   Example:
   Any polynomial with only odd degree terms is an odd function, for example, f(x) = 2x⁷ + 9x⁵ - x. (Note that all the powers of x are odd numbers.)
   Similarly, any polynomial with only even degree terms is an even function. For example, f(x) = 6x⁸ - 6x² - 5. 
   Based on above defination we can call Sine is an odd function, and cosine is even 
   sin (-x) = -sin x, and 
   cos (-x) = cos x.
   These facts follow from the symmetry of the unit circle across the x-axis. The angle -x is the same angle as x except it's on the other side of the x-axis. Flipping a point (x,y) to the other side of the x-axis makes it into (x,-y), so the y-coordinate is negated, that is, the sine is negated, but the x-coordinate remains the same, that is, the cosine is unchanged.
6. Now since in unit circle
   -1 ≤ a ≤ 1
   -1 ≤ b ≤ 1
   It follows that for all x
   -1 ≤ sin(x) ≤ 1
   -1 ≤ cos(x) ≤ 1 Also We know from previous classes,
   a,b are both positive in Ist quadrant i.e 0< x < π/2 It implies that sin is positive and cos is postive
   a is negative and b is positive in IInd quadrant i.e π/2 < x< πIt implies that sin is negative and cos is postive
   a and b both are negative in III quadrant ie. π < x < 3π/2 It implies that sin is negative and cos is negative
   a is positive and b is negative in IV quadrant i,.e 3π/2 < x < 2π It implies that sin is positive and cos is negative
   Similarly sign can be obtained for other functions

7. Domain and Range of trigonimetric functions

   1. y=f(x)= Sin(x)
      Domain : It is defined for all real values of x
      Range : -1 ≤ y ≤ 1
      Period:2π
      It is a odd function
      
   2. y=f(x)= cos(x)
      Domain : It is defined for all real values of x
      Range : -1 ≤ y ≤ 1
      Period:2π
      It is even function
      
   3. ) y=f(x)=tan(x)
      Domain : It is defined for all real values of x except x ≠(2n + 1)(π/2) where n is any 
      Range : All the real numbers
      Period:π
      It is a odd function
      
   4. y=f(x)=cot(x)
      Domain : It is defined for all real values of x except x ≠nπ, where n is any integer
      Range : All the real numbers
      Period:π
      It is a odd function
      
   5. y=f(x)=sec(x)
      Domain : It is defined for all real values of x except x ≠(2n + 1)(π/2) where n is any integer
      Range : (-∞,-1] ∪ [1,∞)
      Period:2π
      It is even function
      
   6. y=f(x)=cosec(x)
      Domain :It is defined for all real values of x except x ≠nπ, where n is any integer
      Range : (-∞,-1] ∪ [1,∞)
      Period:2π
      It is odd function