Group Identity Theory

Introduction about group theory

Group:            

    If a set is defined collectively with the binary o[operation  * then it is called a group. It is defined as the symbol G.If  x and y are the member of a set G and it related to  the binary operation * means then the g is said to be a Group. Here the binary operation should satisfy the certain Axioms.

Axioms of Group theory:

Closure
Associativity of an element
Identity of an element
Inverse of an element.

Properties of group theory


Closure property in group theory:

    If any two elements x ,y  in the group G, then the outcome of the product operation   (x.y) is also present in the group G.

Associativity property in Group theory:

        If any set of elements present in the group G then it satisfies the following condition

                      (x .y).z= x. (y .z).

 Inverse property in group theory:

   For each x is present in the group G , then there exist an element y is also present in  the group  G such that       x . y = y .x =e. here e is defined as an identity element. The outcome of combining element x with y   are not necessitate to acquiesce the same outcome as combining element y with x.

               x. y = y. as It is not always be true.

Identity of an element in the group theory:

If an element x is present in the group G and there exist an another element e in the group G then it satisfies the following,

                 e. x= x . e =x.Here we say that e is an identity element.

Identity of an element is the important axiom of group theory.


Applications of group identity theory:


It is used in the Galois Theory for describe the polynomial roots of symmetries.
The connection between the algebraic field extension and the group theory derived fro m the Galois fundamental theorem.
Another use of group theory is the Algebraic topology. Some of the topological invariants are defined by group.
Group theory is primarily used in the Cryptography and the Algebraic geometry.

Meaning of Integer

Introduction to Meaning of integer:

              The integers are the combination of negative number and non negative numbers. The integers are formed with the help of the natural numbers. The integer set is formed countable infinite set. The integer numbers are denoted with the help of alphabetic letter Z. the natural numbers including the 0. Denoted by 0,1,2,3,….

              Here we are going to understand the meaning of integer by solving some example problems.

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Basic properties of an integers:


Now we see about the some basic properties of an integers.

1) Commutative property of addition of integer:

The commutative property is one of the property of integer addition .it tells that we can add numbers in any order.

For example: -7 + 8 = 8+ (-7)

2) Commutative property of multiplication of integer:

       The commutative property of multiplication is also one of the properties of an onteger that tells  we can multiply numbers in any order doesn’t change result.

For example: -8 x 5 = 5 x (-8).

3) Associative property of addition of integer:

       The associative property of Addition is one of the properties of an integer that tells  we can group together then we get the same result.

For example : (-4 + 3) + 2 = -4 + (3 + 2)

4) Associative property of multiplication of integer:

       The associative property of multiplication is fourth property of an integers that tells we can group together in a product then we get the same answer.

For example : -4(2) x 1 = -4( 2 x 1)

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Integer word problems:


Problem 1:

A team won 4 times as many matches as it lost. If it won 16 matches, how many games did it lose?

Solution:

Step 1: First we assign variable, A team won 4 time s as many matches as it lost.

            Let P be number of matches lost and

            4P is number of matches won.

            It won 16 matches

            So 4P = 16

Step 2: Solve the equation,

            Divide by 4 on both sides.

            We get `(4P)/4` = `16/4`

                              P = `16/4`

                              P = 4.

Answer: The team lost 4 games.

Problem 2:

          John and his friends were selling cookies. They sold 5 more boxes the third week than they did the second. On the fourth week, they doubled the sale of their third week.Altogether, they sold a total of 459 boxes. How many boxes did they sell in the fourth week?

Solution:

Step 1:

        They sold 5 more boxes the third week than they did the second. On the fourth week, they doubled the sale of their third week.

       Assign variables Let,   P be boxes sold in the second week 

                                       P + 5 boxes were sold in the third week 

                                       2(P + 5)  =   boxes sold in the third week

       John sold a total of 444 boxes.

       So, P + P + 5 + 2(P + 5) = 459

       Then remove brackets and combine like terms,

                   P + P + 5 + 2P + 10 = 459

                  4P + 15 = 459

       Subtract -15 on both sides

                4P + 15 – 15 = 459 -15

                    4P = 444

       Divide by 4 on both sides

                     `(4P) / 4` = `444 / 4`

                        P = 111

Step 2:  Plug P = 111 into 2(P+4)

             We get, 2(111 + 4)

                         = 222 + 8

                         = 230

Answer: In the fourth week, they sold 230 boxes.

Meaning of Geometry

Introduction for meaning of geometry:

                    The word “geometry” is derived from grouping of two Greek words “Geo” and “me tron”. The word “Geo” means “earth” and “me tron” means “measurement”. So the subject name is “earth measurement” . It was originally named as “geometry”.

                    Geometry is one of the main branch of mathematics, which involves the study of shapes, line equation, angles, and dimensions, relative position of diagrams etc. The basic diagram of geometry is point, line, square, rectangle, triangle, and circle. The geometry properties and example problems are given below.

Let us see properties and theorems of geometry meaning. And also see solved example problems using geometry meaning. I like to share this Ray Geometry with you all through my article.


Properties and theorems for geometry:


Property 1: If any two points on a plane, there is one and only one line containing them.

Property 2: Two distinct lines cannot have more than one point in common

Property 3: If a line and a point not on it, there is one and only one line that passes through the given point and is parallel to the given line.

Property 4: If the given two lines intersect, then the vertically opposite angles are equal.

Property 5: If a given transversal intersects two parallel lines, then any pair of corresponding angles are equal.

Property 6: If any two sides and the included angle of one triangle are equal to any two sides and the included angle of another triangle, then the two triangles are congruent

Theorem 1:

   The given two triangles are congruent if the three sides of one triangle are equal to the three sides of the other triangle.

             Draw a triangle ABC and a line XY parallel to the side BC (see Figure). Mark the point of intersection of line XY and the side AB as D, and the point of intersection of the line XY and the side AC as E. Since the side AB and the side AC are transversal of the parallel line segments XY and BC,

                               ∠D = ∠B, ∠E = ∠C

           The two triangles ABC and ADE have AAA property. However, they are not congruent since the corresponding sides are not equal. Hence, we conclude that AAA correspondence cannot be a criterion for congruency of triangles.

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Example Problems - Geometry:


Example problem 1:

     Find the supplement of the following angles:

      (i) 70° (ii) 45° (iii) 120° (iv) 155°

Solution:

    Since the sum of supplementary angles is 180°,

          (i) the supplement of 70° is 180° − 70° = 110°.

          (ii) the supplement of 45° is 180° − 45° = 135°.

          (iii) the supplement of 120° is 180° − 120° = 60°.

          (iv) the supplement of 155° is 180° − 155° = 25°.

Example problem 2:

   If the angles of a triangle are in the ratio 4 : 4 : 7, find them.

Solution:

    Let the angles be 4x, 4x, 7x.

    Then 4x + 4x + 7x = 180° or 15x = 180° or x = 12°.

    The angles are 4 × 12°, 4 × 12°, 7 × 12°, or 48°, 48°, 84°.

This is how problems solved using geometry meaning.

Geometry Hexagon

Geometry Hexagon

Hexagon is the two dimensional geometric closed figure with five sides.

Geometry Hexagon interior angle:

The angle that found inside the geometric figure is said to be interior angle of the Hexagon.

Geometry Hexagon Exterior angle:

The angle between any side of the Hexagon and the line extended from the next side is said to be exterior angle of the Hexagon.

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Formula:

Formula for calculating the exterior angle of the Hexagon:

Sum of the exterior angle of the Hexagon is 360 degrees.

Formula for calculating the interior angle of the Hexagon:

Sum of Interior angle of the Hexagon = (n-2) 180 degrees

Here n is the number of the Hexagon.

Regular Hexagons:

The regular Hexagon is the geometric figure  in which all sides are equal in length and all the angles are equal in degrees.

Interior angle of the regular Hexagon is `(((n-2) 180)/n)`

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Model Problems:

1.Find the sum of interior angle and each interior angle of the regular geometry Hexagon?

Solution:

To find the sum of the interior angle:

Here

Sum of the interior angle of Hexagon = (n-2) 180 degrees

Here n = 6 sides

= (6-2) 180

= (4) 180

Sum of the interior angle of Hexagon =720 degrees.

To find the each interior angle:

Here

Interior angle of the regular Hexagon =` (((n-2) 180)/n)`

= (((6-2)180)/5)

= (720/6)

Interior angle of the regular Hexagon = 120 degrees.

2. Find sixth interior angle of the Geometry Hexagon when first five angles are 108, 110, 106, 102, 104 degrees respectively?

Solution:

We know that sum of the interior angle of the Hexagon = 720 degrees

That is (angle 1+angle2+angle 3+angle 4+angle 5 +angle 6) = 720

(108+110+106+102+104+x) = 720

(530+x) = 720

x = 720-530

Sixth interior angle of the Hexagon x = 190 degrees

3. Find first interior angle of the Geometry Hexagon when last five angles are 108, 110, 108, 106, 104 degrees respectively?

Solution:

We know that sum of the interior angle of the Hexagon = 720 degrees

That is (angle 1+angle2+angle 3+angle 4+angle 5 +angle 6) = 720

(x+108+110+108+106+104) = 720

(x+536) = 720

x = 720-536

Sixth interior angle of the Hexagon x = 184 degrees

Fractionation Method

Introduction Fraction Method:

A fraction is a value that shows the number of equal parts taken of a whole quantity or unit. The denominator of a fraction is the number that shows how many equal parts are in the whole quantity. The numerator of a fraction is the number that shows how many equal parts of the, whose are taken.

The numerator and denominator are called the term of the fraction,

3 -(Numerator)
-------------------------
4-(Denominator)

A improper fraction is a fraction in which the numerator is larger than equal to the denominator, as in 3 / 2, 5 / 4, 11 / 8. A mixed number is a number composed of a whole number and a fraction, as examples 3 7/8, 7 1/2

A complex fraction is a fraction in which one or both of the terms are fraction or mixed number, as in example ¾ / 6.

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Fraction methods - Addition and Subtraction:


Steps for fraction method - addition:

To add algebraic fraction, follow these steps:

Write the given fraction and common denominator.
Added the numerators value.
Solution for the problem
Example:

`= (1 / 2) + (5 / 2)`

`= (1 + 5) / 2`

`= 6 / 2`

= 3

Steps for fraction method - subtraction:

To subtraction algebraic fraction, follow these steps:

Write the given fraction and common denominator.
Subtracted the numerators value.
Solution for the problem
Example:

` = (7 / 3) - (4 / 3)`

` = (7 - 3) / 3`

= `4 / 3`

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Fraction methods – Multiplication and Division:


Steps for fraction method - multiplication:

To multiply algebraic fraction, follow these steps:

Write the given fraction and cancel any common factors.
Multiply the numerators.
Multiply the denominators.

Example:

= `(3 / 7)xx (4 / 5)`

= `(3xx 4) / (7 xx5)`

= ` 12 / 35`

Steps for fraction method division:

To divide algebraic fraction, follow these steps:

Write the given fractions.
Change the division sign to a multiplication sign and invert the second fraction.
Write the given fraction and cancel any common factors.
Multiply the numerators.
Multiply the denominators.
Example:

= `( 1 / 6) / (3 / 4)` (divisor)

= `(1 /6)xx (4 / 3)`

=` (4 xx1) / (6xx 3)`

= `4 / 18`

= `2 / 9`

Solve One-to-one Function

Solve One-to-One Function

One-to-one function is a function, in which every element of the range of the function is corresponds to exactly one element of the domain of the function. One-to-one function is often written as 1–1 function.

Example:

Function: y = f(x) is a function, if it passes only the vertical line test.

One-to-function: y = f(x) is a one-to-one function, if it passes both the horizontal line test and the vertical line test.


Solve One-to-One Function – Example Problems


See these solved problems on one-to-one function.

Example 1: Show that the function f(x) = 4(x - 6)3 + 9 is one-to-one function.

Solution:

Let (x) = f(y)

4(x - 6)2 + 9 = 4(y - 6)3 + 9

Add -9 to both sides

4(x - 6)3 + 9 - 9 = 4(y - 6)3 + 9 - 9

4(x - 6)3 = 4(y - 6)3

Divide both sides by 4

(x - 6)3 = (y - 6)3

The above equation leads to two other equations

(x - 6) = (y - 6)

x = y

Therefore given function f(x) = 4(x - 6)2 + 9 is one-to-one function.

Example 2: Show that the rational function f(x) = `10 / (12x + 15)` is one-to-one function.

Solution:

Let f(x) = f(y)

`10 / (12x + 15)` = `10 / (12y + 15)`

Multiply both sides (12x + 15)(12y + 15) and simplify

12y + 15 = 12x + 15

Add -15 to both sides

12y = 12x

Divide both sides by 12

y = x

Therefore the given function f(x) = `10 / (12x + 15)` is one-to-one function.

Example 3: Show that the function f(x) = 15x + 18, is one-to-one function.

Solution:

Let f(x) = f(y) and show that this leads to x = y

15x + 18 = 15y + 18

Add -18 to both sides

15x = 15y

Divide both sides by 15

x = y

Therefore the given function f(x) = 15x + 18 is one-to-one.

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Solve One-to-One Function – Practice Problems


Solve these following practice problems

Problem 1: Show that the rational function f(x) = `7 / (11x + 6) ` is one-to-one function.

Problem 2: Show that the function f(x) = 5(x - 11)3 + 21 is one-to-one function.

Problem 3: Show that function f(x) = 19x + 4 is one-to-one function.

Exterior Angle Solve Online

Introduction - Exterior angle solve online:

In this article, we shall discuss about Exterior angle solve online. Online helps students to share their views as well as gather notes regarding their subject. In geometry, shapes play an important part. There are various kinds of shapes. Any plane figure which is formed at the vertex where two lines intersect are called angles. The angles formed at the outer part of each vertex are called as exterior angles. There are two methods to find the exterior angles.

Method 1:

Exterior angle = `360/n` ,

where n is the number of sides of the polygon.

Method 2:

Exterior angle = 180 - interior angle.

Now we shall solve some example problems regarding exterior angle solve online.


Example Problems - exterior angle solve online:


Example 1:

Solve for the exterior angle of a polygon whose interior angle is 108°. Also determine the number of sides of the polygon.

Solution:

Given, The interior angle of a polygon is 108°.

When the interior angle is given, the exterior angle of a polygon can be calculated by using the formula,

Exterior angle = 180 - Interior angle

Exterior angle = 180 - 108°

= 72°

The exterior angle of the polygon is 72°.

Now the number of sides of the polygon can be determined by using the formula,

Exterior angle = `360/n`

where n is the number of sides of the polygon.

We know that the exterior angle of the given polygon is 72°

72 = `360 / n`

Multiply by n on both sides,

`72 n = n[360/n]`

`72 n = 360`

Divide by 72 on both sides

`(72n) / 72 = 360/72`

`n = 5`

Therefore the number of sides of the polygon is 5.

Since n = 5, the name of the polygon is pentagon.

Example 2:

Solve for the exterior angle of a polygon whose number of sides is 8.

Solution:

When the number of sides of the polygon is given, the exterior angle can be calculated by using the formula,

Exterior angle = `360/n`

where n is the number of sides of the polygon.

By substituting n = 8 in the formula, we can obtain the exterior angle of the polygon.

Exterior angle = `360/8`

= 45°

Since n = 8, the name of the polygon is octagon.

Hence the exterior angle of octagon is 45°.

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Practice Problems - exterior angle solve online:


Problem 1:

Solve for the exterior angle of a polygon whose interior angle is 120°. Also determine the number of sides of the polygon.

Answer:

Exterior angle = 60°

Number of sides = 6

Problem 2:

Solve for the exterior angle of a polygon whose number of sides is 10.

Answer:

Exterior angle = 36°.