Set Theory Proper Subset

Introduction

We can define set as an ordered group of objects,with no repetition of elements.The famous mathematician George cantor is widely known as the founder of set theory.

A set is a grouped under some common properties of the elements of the set.

Ex:set of numbers,letters, vegetables etc..

W e can represent a set in mainly 2 ways.

1.Intensional

2.Extensional

One is the way of  describing the common nature of the elements of the set and other is the way of writing it

One by one

Ex: set of all natural numbers

{1,2,3,4….}.Here we can see the difference of the 2 methods.

A set is said to be a subset of another set if all of its elements belongs to the other set.

A set is a subset of itself.As it contains all of its elements.

Proper Subset and Subset

A subset and propersubset is different.

As we told previously a set is a subset of itself.But the proper subset of a set  w ill be having atleast 1 element less than the original set.

Ex: A={1,2,3,4,5}

B={2,3}

Here B is a proper subset of the set A.,Because A contains all elements of B and the number of elements are less than the elements in A.

Now, see C={1,2,7} and X={4}

Here C cant be a proper subset of A,as it contain the elment 7,which is not present in A.

But the set X is a proper subset,as it  satisfy the 2 conditions of the proper subsets.

Venn diagram:

The  Idea of subsets can be easily got by the help of venn diagrams.We can see How a subset of a element is represented in Venn diagrams

Here the set A is the subset of B.In mathematical notation we can write it as

A`sub` B ,and B is the super set of A.

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Proper Subset Problems

Practice problems:

1.If A={a,c,e,g,I,k,m}  write 3 proper subsets of the set

Answer:

1.A1={a,c}

2.B1={}

3.A2= ={a,c,e,g,I,k}

Here just note that the null set is the subset of all sets.

2.Write all proper subsets of Z={2,3,4}

Answer:

Z1={}

Z2={2}

Z3={3}

Z4={4}

Z5={2,3}

Z6={2,4}

Z7={3,4}

So here we can see that the set of all proper subsets of a set is 2 x  -1

Where x is the number of elements of the set.

Note:We can see this relation is true for all sets.We substracted 1 here because ,we were asked to find the set of all proper subsets.

If  it was for subsets then 2x is the formula.

Sum of Factors of a Number

Introduction :

A number that be multiply together to get another number is called as factors. The sum of factors of the number is nothing but the adding the factors of the number. First step for the sum of factors of the number is to be factor the number and then added the factors. In this article, we see about the sum of factors of a number.

Sum of Factors of a Number - Example Problems:

Example 1:

Find the sum of factors of the number 56.

Solution:

Step 1: The factors of the number 56 is divided by 1, 2, 4, 7, 8, 14, 28, and 56

Step 2: Sum of factoring a number 56.

1 + 2 + 4 + 7 + 8 + 14 + 28 + 56 = 120

Answer: 120

Example 2:

Find the sum of factors of the number 78

Solution:

Step 1: Factoring of the number 78 can be dividing by 1, 2, 3, 6, 13, 26, 39, and 78

Step 2:  Sum of factoring a number 78

1 + 2 +3 + 6 + 13 + 26 + 39 + 78 = 168

Answer: 168

Example 3:

Find the sum of factoring the number 12

Solution:

Step 1: factoring the number 12 is divided by the number 1, 2, 3, 4, 6, and 12.

Step 2:  Sum of factoring a number 12

1 + 2 + 3 + 4 + 6 + 12 = 28

Answer: 28

Example 4:

Find the sum of factors of the number 100

Solution:

Step 1: Factoring the number is 100 is divided by 1, 2, 4, 5, 10, 20, 25, 50, and 100.

Step 2: Add the factors of the number 100.

1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 + 100 = 217

Answer: 217

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Practice Problem – Sum of Factors of the Number:

Problem 1:

Find the sum of factors of the number 8

Solution: 15

Problem 2:

Find the sum of factors of the number 15

Solution: 24

Distributive Property Exponents

Introduction of Distributive Property Exponents:
Exponent is said to be the number to the power. Example: Am, here the number A is the base number and the power of the number A is m which is called exponent.

Here we see about the distributive Properties of Exponents.

The general form of the distributive Properties of Exponents:

(AB)m = Am Bm

Proof - Distributive Property Exponents:

Let take the example of (PQ)2

To prove (PQ)2 = P2 * Q2

(PQ)2 = PQ * PQ

= P * Q * P * Q

= P * P * Q * Q

= P2 * Q2

We proved the distributive property exponents.

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Example Problem - Distributive Property Exponents:

Case 1: Base - Number and Exponent - Integer:

Example 1:

What is the value of the expression (2242)2?

Solution:

(2242)2 = 22 * 2 42 * 2

= 24 44

Answer: (2242)2 = 24 44

Case 2: Base – Variable and Exponents – Integer:

Example 2:

What is the value of the expression (A2B3)2?

Solution:

(A2B3)2 = A2 * 2 B3 * 2

= A4 B6

Answer: (A2B3)2 = A4 B6

Case 3: Base – Variable and Exponents – Negative and Positive fraction:

Example 3:

What is the value of the expression `(A^(3/2) . B ^ (-1/4))^2` ?

Solution:

`(A^(3/2) . B ^ (-1/4))^2`=  `A^(3/2 * 2) . B ^ (-1/4 * 2)`

=  `A^(3) . B ^ (-1/2)`

Answer:  `(A^(3/2) . B ^ (-1/4))^2` =  `A^(3) . B ^ (-1/2)`

Case 4: Base – Variable and Exponents – Positive decimals:

Example 4:

What is the value of the expression (A0.5 B1.3)3?

Solution:

(A0.5B1.3)3 = A0.5 * 3 B1.3 * 3

= A1.5 B3.9

Answer: (A0.5B1.3)3 = A1.5 B3.9

Case 5: Base – Variable expressed in fraction and Exponents – Integer

Example 5:

What is the value of the expression `((A^2)/(B^3))^2` ?

Solution:

`((A^2)/(B^3))^2` = `A^(2 * 2)/(B^(3 * 2))`

= `A^(4)/(B^(6))`

Answer: `((A^2)/(B^3))^2`= `A^(4)/(B^(6))`

Case 6: Using Exponent – zero

Example 6:

What is the value of the expression (A0B)2?

Solution:

(A0B)2 = A0 * 2 B1 * 2

= A0 B2

= 1 * B2 = B2

Answer: (A0B)2 = B2

Practice Problem - Distributive Property Exponents:

Problem 1:

What is the value of the expression (X2 A)2?

Answer: X4A2

Problem 2:

What is the value of the expression (P0Q2)1/2?

Answer: Q

Rulers that Measure Angles

Introduction to rulers that measure angles:

To measure an angle  we need a ruler called a protractor. The protractor is the common ruler which is used to measure an angle. It is very simple to measure an angle. In this way it will be very easy for us to measure the angles. This is the common way used allover the world for measuring an angle. There are four different types of angles that can be measured by using the protractor. Here we are going to see what is the way to measure angles through the protractor.

What is a Protractor?

This ruler name is called protractor, This is the ruler used to measure angles. There will be angles marked from 0 to 180.  When we want to measure an angle we have to keep the bottom line of the protractor and the angle that we have to measure on a same straight line. And then we have to see on which degree it projects. This will help us to figure out the angle. Protractor angle measurement will be very easy to find the angle.

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Ways to Measure Angles:

Measuring an angle is very simple, When the angle is top of the line we have to measure it by keeping the protractor facing top.This is one of the ways of measuring an angle.

An angle is drawn and that angle is measured as 60 degree. we can see that the protractor is kept in the base line A and then we can see the projected angle B at the top of the protractor. So we can see the marking 60 degree in the protractor. So the angle given is 90 degree. This is how we figure out an angle using the protractor.

There are four different types of angles that can be figured out while measuring angles using protractor:

When we figure an angle, if it measures less than 90o (<90o acute="acute" an="an" angle.="angle." as="as" br="br" is="is" it="it" known="known" then="then">When we figure an angle, if it measures exactly 90o is known as a right angle.
When we figure an angle, if it measures greater than 90o and less than 180o is known as the obtuse angle.
When we figure an angle, if it measures exactly 180o is known as straight angle. So the straight angle looks line a straight line.

Substitution Method Calculus

Introduction
In this article we are going to discuss about the substitution method calculus problem concept. The process of short form the calculus corresponding to over screening the significant concept and problem in substitution method calculus are referred as review calculus. This article helps to improve the knowledge for using substitution method calculus problem and below the problems are helping toll for the exam. Go ahead of the test using review to this article. Substitution method calculus problem solutions also show below.

Level One Example Substitution Method Calculus Problems:-
Substitution method calculus problem 1:-

Integrals evaluated using the method of substitution:

`int sin (3x+5) dx`

Solution:-

Step 1:-

Let` u = 3x + 5`

Step 2:-

Then `du = 3dx`

Step 3:-

`int sin (3x + 5) dx = int sin (u)*(du)/ (3)`

`=1/3 int sin (u) du`

Step 4:-

`= 1/3 [- cos (u)] + C`

Step 5:-

`= 1/3 [- cos (3x+5)] + C`

Substitution method calculus problem 2:-

Integrals evaluated using the method of substitution:

`int (sqrt (7x-3)) dx`

Solution:-

Step 1:-

Let `u = 7x -3`

Step 1:-

Then `du = 7 dx`

Step 2:-

`int (sqrt (7x-3)) dx = int u^ (1/2) (du)/ (7)`

`= 1/7 int u^ (1/2) (du)`

Step 3:-

`= 1/7 u^ (3/2)/ (3/2) + C`

Step 4:-

`= 2/21 (7x - 3) ^ (3/2) +C.`

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Level Two Example Substitution Method Calculus Problems:-
Substitution method calculus problem 1:-

Integrals evaluated using the method of substitution:

`int 9 x e^ (5 - 3x^2) dx`

Solution:-

Step 1:-

Let` u = 5 - 3 x^2`

Step 2:-

Then` du = -6 x dx.`

Step 3:-

`int9 x e^ (5-3x^2) dx = int 9 e^ (5-3x^2) x dx`

`= int 9 e^u (du)/ (-6)`

Step 4:-

`= (-3)/ (2) int e ^u du`

`= (-3)/ (2) e^u + C`

Step 5:-

`= (-3)/ (2) e^ (5-3x^2) + C`

Substitution method calculus problem 2:-

Integrals evaluated using the method of substitution:

`int sin (5x) cos (5x) dx`

Solution:-

Step 1:-

Let `u = sin (5x)`

Step 2:-

Then `du = cos (5x) 5 dx.`

Step 3:-

`int sin (5x) cos (5x) dx = int u (du)/ (5)`

`= 1/5 int u du`

Step 4:-

`= 1/5 (u^2)/ (2) + C`

`= (1)/ (10) sin^2 (5x) + C.`

Ray Line Segment

Introduction For Ray Line Segment:

Ray:
A ray has only starting points but it doesn't have an any ending points. A ray is a part of a line and  We draw only a parts of a line and marked arrow-head at it  two ends to represent that rays extends endlessly in both directions.

Line segment:
A line segment is a joined piece of a line. Line segment  has two endpoints and it is named at its endpoints. Sometimes, the symbol – written on top of two letters is used to denote the segment.

Ray:

A rays of light  which originates from the point in the sun or from torch light   and extends endlessly in one direction. Thus we can say that a rays starts from constant point and its extended endlessly in one direction.

Sun emitting  the rays of light in all directions. We can see,the rays of light from the torch light. These shows idea for rays.
The initial point is the vertex for  the angle and the two rays are called arms the of the angle.
An angle expressed by an arc marked between the arms of the angle Two rays with the same initial point form an angle. Common and initial point is said to be the vertex and the two rays are called the arms of the angle.

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Concurrent Ray Line Segment :

Line segment:
A line segment is a part of a line which  consisting of two end points on it.

Draw the line segment as  l. Mark two points on the  line l as  A and B.
The portion start from  the line l from A to B is called  line segment AB. line is denoted by AB or BA.
It is named as line segment AB or line segment  BA.
The line segment AB and  line segment BA are the similar and they are equal in measure.

Mark a point P on the paper. Draw a line segment  ‘l’  which passing through the point P.
Draw another line segment ‘m’ through the point P. Continue drawing line segment process .
Many lines can be drawn through this point P
All the line segment passing through the same point P are called concurrent line segment.
The point P is represented as point of concurrence.
More than  three  lines passing through  same point are called concurrent lines.
The point which line pass through is  the point of concurrence.

Perfect Squares Chart

Perfect squares chart :

 Perfect squares are defined as expressions or numbers that can be factored into two equal expressions or numbers.

For example,

x² + 2xy + y² = ( x + y) * ( x + y) , so  (x² + 2xy +  y²) is a perfect square of (x + y)

4 = 2*2 , so 4 is a perfect square of 2.

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Properties of Perfect Squares Chart

All perfect squares are non-negative ( at least in the real numbers' set).

• Square of 0 is 0.
• Squares of perfect numbers are the same as the squares of their positive counterparts,  

 For example           -1*-1 = 1*1 = 1,  (-2)²= 2²= 4

Difference between two perfect squares

Difference between a perfect square and the next perfect square is given by :

Square of a number n = n²

The next perfect square is square of the next number, that is, square of n + 1, so ( n + 1) ² = n² + 2n + 1

Difference between the two numbers =  2n + 1

So, if n² is a perfect square, the next perfect square will be n² + ( 2n + 1).

Example : 

100 = 10² is a perfect square.

Here, n = 10

So, the next perfect square will be 100+ 2*10 + 1 = 121 = 11²

Uses of Perfect Squares Chart

Included below is a chart of perfect squares of numbers from 1 to 25

Number n	Number Square n2
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625

Remember: 

• A square number can only end with digits 00,1,4,6,9, or 25
• Squares of even numbers are even, since (2n)² = 4n².
• Squares of odd numbers are odd, since (2n + 1)² = 4(n² + n) + 1.
• It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.

Uses of Perfect Squares :

Squaring is used in statistics in determining the standard deviation of a set of values. The deviation of each value  from the mean  of the set is defined as the difference . These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the variance and its square root is the standard deviation. In finance, the volatility of a financial instrument is the standard deviation of its values.