Sets and relations

Sets and relations:

Introduction about sets and relations:

A group of elements is called a set when the elements in the group are distinct. A characteristic of two objects is called Relation
.
Example of a Set:

• The collection of all natural numbers
• The collection of all equilateral triangles in a plane.

Let us learn more about sets and relations in this chapter.

A relation is a set of ordered pairs(x,y) , where the first component of the ordered pairs are the input values and the second component are the output values.

Set of all Input Values is known as Domain of the given Relation.

Similarly, set of all Output Values is known as Range of the given Relation.

Mathematically,

A relation R defined by ( , {(x,y):x }) ,is the set of ordered pairs of all Input Values (x) and all Output Values(y) such that each element x is related to the corresponding element y.

Here,

x=Domain of the Relation and

y=Range of the Relation

iff no values of x and y is repeated.

x is called Independent variable and y is called Dependent variable because its value depends on the x-value chosen.The most commonly asked question is how do we express a relation,we come across this question very frequently. Relation scan be represented in any of the following forms:-

(i) Roster form

This method is also known as Tabular method. In this method, a set is represented by writing all the elements of the set, separated by commas and are enclosed withinbrackets { }.

For e.g. D = {Sunday, Monday, Tuesday,, Wednesday, Thursday, Saturday}.

Lets take some examples to show the relation in various forms:-

(ii) Set builder form Set builder notation has the form {x : f(x)} (some write {x | f(x)}, using the vertical bar instead of the colon), denoting the set of all individuals in the universe of discourse satisfying the formula f(x), that is, the set whose members are every individual x such that f(x) is true.For e.g. is the set of all positive real numbers.

(iii) By tables

Table representation of the RelationInputOutputa3c2e9
(iv) Arrow diagram :-

(v) By graphs:-

Hope you like the above example of Sets and Relations. .Please leave your comments, if you have any doubts.

Laws of Exponent

Laws of Exponent:

Introduction to Exponents:

Exponent is a math function symbolize by 2 parts ,

is the base.
is the exponent.

The exponent says the number of times the base must be multiplied by it self, Exponent is the part of algebra and they have defined properties, with the help of it we can simplify the given exponent expressions.

Now lets see the laws of exponents.

The laws of exponents are used for combining exponents of numbers. Exponents is a number raised to another number, it is denoted as, a ⁿ, here, n is known as the exponent of the nth power of a.

Pythagoras Theorem

Pythagoras Theorem:

The Pythagoras Theorem is the theorem which is developed by a Greek mathematician named Pythagoras, for finding the sides of any right angle triangle. It is also used to verify the triangle is Right angle triangle or not. According to him, the sum of square of two smaller sides is equal to the square of largest side. It is given by the formulae,

According to Pythagorean Theorem: c² = a² + b².

The longest side of a right angle triangle is always directly across from the 90 degree angle is the hypotenuse. The other two sides are called as legs.

Students may have heard of the Pythagoras Theorem and the problems related to the theorem, but are uncertain about how to solve these kinds of problems or why they would be important in their everyday lives.

To start with, the student should understand that two lines intersect in a point, called a "vertex" and the circular span between the lines is called an angle.A 90 degree angle is the right angle.


Practice Problems on Pythagoras Theorem Learning

Learn to find the missing measure of leg of the right triangle if hypotenuse is 10 cm and right side is 6cm.

The figure is given above for reference.

Hope you like the above example of Pythagoras Theorem.
Please leave your comments, if you have any doubts.

Whole numbers

Whole numbers:

The set of whole numbers is the set of natural numbers along with zero. so W = the set of whole numbers = 0,1,2,3,............

so Zero is the least number of the set of Whole numbers.

As the whole numbers is an infinite set we cant determine the highest number of this set.

The set of Whole numbers is a subset of Rational numbers.The best way to learn about whole numbers is to understand its properties,the properties of whole numbers are,it is advisable to use a number line inorder to understand the properties of whole numbers.

1. Number 3 <>

2. There is no whole number to the left of zero on the number line.So zero is the smaller number than each of the numbers to its right on the number line.That means 0 is the smallest or least of the whole numbers.

3. A whole number which is greater than a given whole number by 1 is said to be a successive whole number. 1 is the successive whole number to 0.Every whole number has one successor.

4. There is no whole number left to zero.hence 0 is not a successive whole number of any whole.

Hope you like the above example of Whole numbers.
Please leave your comments, if you have any doubts.

Pythagorean theorem:

Pythagorean theorem:

Introduction:

Let us learn about Pythagorean theorem,
Pythagorean theorem states that square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other sides.Consider the right angle triangle, a and b are the two smaller sides and c is the hypotenuse, so if we square a and b and add the both value we get the same answer as if we square c (c²)

This theorem can also be written as an equation relating the lengths of the sides a, b and c : where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.In the figure given above we can see the AC is called a Hypotenuse of the triangle and AB is called the opposite side of the triangle and BC is called the adjacent side of the triangle. This is the Pythagorean Theorem. It generally frames as

(Hypotenuse) ²= (Opposite side) ² + (Adjacent side) ²

Now that we have learned about the meaning of the pythagorean theorem let us also look at a few examples of thePythagorean Theorem,

Let a = 3 cm, b = 4 cm and c = 5cm.

By Pythagorean Theorem,

Thus 32+ 42= 9 +16 = 25 = 52

In a triangle ABC, AB = 5 cm, BC = 3 cm and AC = 4 cm

Examples 2:

Let a = 2.5 cm, b = 6 cm and c = 6.5cm

By Pythagorean Theorem,

Thus (2.5)2+ 62= 6.25 +36 = (6.5)2

In a triangle ABC, AB = 2.5 cm, BC = 6 cm and AC = 6.5 cm.

Hope you like the above example of Pythagorean Theorem.Please leave your comments, if you have any doubts.

Trigonometric Ratios

Trigonometric Ratios

Before we learn about the Trigonometric Ratios let us learn about the meaning of Trigonometry in brief.

Trigonometry is a word consisting of three Greek words " Tri" means three, "Gon" means side, and "Metron" means measure. Thus, trigonometry is a study related to the measures of sides and angles of a triangle. Trigonometry is mainly used by captains of ships to find the direction and distance of islands and light houses from sea. Trigonometry is also used in astronomy, geography and engineering.
Trigonometric Functions and ratios can be studied with the help of examples:
The trigonometry radio co-ordinate plane, consider a point A on the +ve side of x-axis. The trigonometric ratios (circular functions) are defined as follows:

*

The sine of the angle [theta] is defined as the ratio r/r it is denoted by sin [theta]
*

sin [theta] =y/r ; cosecant values at [theta] =r/y = cosec [theta] ; y ≠ 0 and cos [theta] =x/r ; secant values at [theta] =r/x = sec [theta] ; x ≠ 0
*

tan [theta] =y/x ; cotangent values at [theta] =x/y = cot [theta] ; y ≠ 0

Let us learn some problems related to Trigonometric Ratios:

In any right-angled triangle ΔABC,That is given above,

let angle B = 90 o and angle C = Θ.

Line segment AC is the hypotenuse.

With reference to angle C, we can say that,

Line segment AB is the opposite side of

Line segment BC is the adjacent side of

Therefore, trigonometric ratios are given as,




Hope you like the above example of Trigonometric Ratios.Please leave your comments, if you have any doubts.

Arithmetic Progressions

Arithmetic Progressions

In mathematics, an Arithmetic Progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference 2.We usually come across many problems related to arithmetic progression,and we can solve these problems once we learn to identify the sequences.These are basically a set of numbers arranged in a definite order according to some definite rule is called a sequence.A sequence is a function whose domain is the set N of natural numbers.

Indicated sum of the terms in a sequence is called a series.The result of performing the additions is the sum of the series.It is easy to identify the sequence in arithmetic progression as soon as we look at the problem,let us now look at the examples of arithmetic progression.

Quantities are said to be in Arithmetic progression when they increase or decrease by a common difference.
Examples:

Each one of the following series form an Arithmetic progression
i) 1, 3, 5, 7, …

ii) 3, 7, 11, 15, …

iii) 15, 12, 9, …

iv) x, x - d, x - 2d, .....

The common difference is found by subtracting any term of the series from the immediate succeeding term.
In the above example, common difference in the first is 2, in the second it is 4, in the third it is -3, in the fourth it is -d and in the fifth it is d.

The general form of an A.P. is as follows:
a = first term, d = common difference, then A.P. is a, a+d, a+2d, a+3d,.....

Hope you like the above example of Arithmetic progression.
Please leave your comments, if you have any doubts.