Progression definition

Progression definition – it means progress, growth, gain, move forward or instantaneous changes in forward direction. Electromagnetic waves travel forward. It means they are progressing. Plant is growing, it means it is progressing.
Patient health is improving or progressing. The student is progressing; it means that the student is passing his examinations from lower class to higher class. It is the progress in positive or negative direction. Speed may be increased or decreased. It may have acceleration or retardation both.

In mathematics there are three types of progressions.
Arithmetic progression(A.P) :- this take place according to a certain formula, for example
a, a+d, a+2d, a+3d, …………………a+(n-1)d
where a=initial or first term,
d= common difference between the two terms.
n= number of terms

Example1: 1,3,5,7………………..nth term
Sequence of terms is arranged in such a manner so that the difference of any two successive terms of the sequence remains constant

Geometric progression (G.P) :-The rule or formula for G.P is as under.
a,ar,ar²,ar³,…………………..ar^(n-1)
where a=initial or first term,
r= common ratio between the two terms.
n= number of terms
 Sequence of terms is arranged in such a manner so that the ratio of any two successive terms of the sequence remains constant
Harmonic Progression:- a H.P. is the reciprocals of an arithmetic progression. The form of HP is
1/a1,1/a2……………………
a1, a2, a3… is called an HP if  1/a1,1/a2..is an HP.
If a, b, c are in HP, then b is the harmonic mean between a and c.
In this case, b = 2ac/(a+c)
Examples are
12, 6, 4, 3,  , 2, … ,
10, 30, -30, -10, -6, -  , … ,

Progression In other fields:
Astrological:- It is used in Horoscope astrology for  forecast of future happenings
Horizontal:- the gradual movement from left to right or right to left of  a line .
Age:- the process of modification of a photograph of a person show  the  aging effect.
Tax: is a tax in which the amount of tax id directly proportional to the amount of income tax rate is increased as the income amount is increased.
Semantic:- It is the evolution of used word
Color:- the whole ranges of color for which the values changes smoothly through hues, and saturation, along with the illumination, or the combination of all these three factors.
Educational:- It is the progress of the  individual from lower classes to higher class.
Age:- the process of modification of a photograph of a person show  the  aging effect.
Tracking purpose:-It is used in video games to check the performance of the individuals.

Empty set

An empty set, as obviously indicates, is a set which is empty. In other words in an empty-set there are no elements. Sometimes it is also referred as null set. An empty set symbol can be any of the followings.      { } , (/),Ø. Of all these, the last symbol is very prominent.

It may be noted that it is not same as the Greek letter F. There is an interesting point in referring about empty-set. A set when empty is in a way common to all the sets that we think of. Therefore a set with no elements is better referred as ‘the empty set’, the word ‘the’ is more apt than the word ‘an’.

The null set, apart from the fact of having no elements has other properties too. The empty set can be a subset of any set. This may be a bit difficult to understand but a Venn diagram will be helpful to understand clearly. The union of the null set with any set A is the set A itself. It is obvious because we add only a 0 to the number of elements of A.

The intersection of the null set with any set A is again the null set. It is very clear that the intersection set cannot have any elements and it has to be empty. There can be only one subset of the null set which is the null set itself.

Now let us discuss about power set of the empty set. Before that let us see what a power set is. A power set of a set A is the set of all possible subsets of set A, including the null set. For example, if A is {a, b, c}, then all the possible subsets are { }, {a},{b},{c},{a, b},{b, c},{c, a},{a, b, c}. Therefore, the power set AP(A) is defined as {{ }, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}. Thus we find there are 8 subsets for a set A having three elements.

That is, 23 subsets for 3 elements and in set language the power set P(A) in this case can be expressed as P(A) = 23.  Extending this concept in general for a set A having ‘n’ number of elements, P(Ø) = 2n. Therefore, in case of a null set the power set of the null set is P(A) = 20 = 1. That is the null set itself. In other words, P(Ø) = { }.

Learning About the Field of Vectors

The quantities in mathematics can have different features. The features can be magnitude and sometimes direction as well. All the quantities do not have the same feature. This difference in features can be the difference between the quantities. The quantities that have only magnitude are known to be scalar quantities and the quantities that have both the magnitude and a direction can be called vectors or carriers.

This is the basic difference between the two types of quantities. This difference must be understood to understand the difference between the two types of quantities. This difference can be very helpful in understanding the difference between the quantities. The vector space over a field can be understood only when the concept of carriers is clear and understood. It is part of a mathematical in which the carriers which can be multiplied with a scalar.

The operation of addition is also possible within this space. The scalars can be any numbers. Sometimes they can also be real numbers or can also be complex numbers. The complex numbers have an imaginary part attached to them. The flux of vector field is nothing but a flow property of carriers. This property is concerned with a unit area.

The flow is per unit area. This concept is very important in the field of physics also. Both mathematics and physics are related to each other. So, this concept appears in both physics and mathematics. The knowledge of physics will help in understanding of the concept. The mathematical plot vector field will explain the concept.

The pictorial representation of the concepts will help in the understanding of the concepts. So, the pictorial representations are always helpful.

The flow of a vector field is related to the flux of the field. These both concepts are interrelated and the understanding of one concept will help in the understanding of the other concept. The plot can be shown in the Cartesian plane using the ‘x’ and ‘y’ axes.

The vector field graph shows the orientation of the various carriers in the Cartesian plane. The carriers are shown aligned in the different directions. So, the orientation of the carriers can be easily understood with the help of this graph.

This graph shows the different directions associated with the carriers. The carriers are shown oriented in different directions and this can be understood from the graph. So, the graph can be very useful.

Introduction to Linear Algebra

Linear Algebra is one of the branches of Mathematics.  These helps with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are found in modern math thus, it is widely used in both abstract algebra and functional analysis.
It is also found in solving problems in analytic geometry.
 Its application is found extensively  the natural sciences and the social sciences.  This forms an intro to Linear Algebra.

 The mathematical approach of solving for variables in equations is Algebra.  
It does the manipulation of these variables using mathematical structures.
It is like taking algebra to another level by manipulating these variables using mathematical structures. This also helps to write the entire system of equations in the form of matrices. When expressed in the form of matrices this to study in depth of the individual parts of matrix which is known as vertex. A vertex is simply a representation of a coordinate.

Linear algebra is a branch of mathematics.  The creation of analytic geometry led to  the system of linear equations  acquire a new significance.
Linear-algebra is important for both pure mathematics and applied mathematics.    
Linear form of algebra when combined with calculus facilitates the solution of linear systems of differential equations.

Analytic geometry, engineering, physics, natural sciences, computer science, and the social sciences   use the techniques of Linear-algebra.
Linear mathematical models are sometimes used to approximate Non-linear mathematical models because linear algebra is such a well-developed theory.
The real world applications need Linear-Algebra.  Algebra is all about linear system of equations and their solutions Linear-Algebra usually consists of the linear set of equations as well as their transformations on it.  Linear Equations  consists of  topics  like  Linear Equations , Matrices, Determinants, Complex numbers, Second degree equations, Eigen values ,Linear Transformations.

Linear algebra help to mathematics: It is a useful branch of mathematics.   Below listed is some of the applications of Linear form of Algebra.  Constructing curves, Least square approximation, traffic flow, Electrical circuits, Determinants, Graph theory, Cryptography.
It   is used to draw graphs. Most equations of linear algebra will   be a straight line. For example draw a graph for a vehicle travelling at a constant speed at various time intervals.
The graph helps in determining the    unknown variable that is the distance by plotting in on the graph.  This can also be used for   a multitude of different functions that is a ready tool for  lots of different real life functions.

Learning to Solve Algebraic Equations

There are various disciplines in the field of mathematics. It is a very interesting subject to learn and can be fun. To simplify algebraic equations, the concept of equations has to be clear. In equations there are both constants and variables present.

There is difference between both these. The value of constants almost remains same throughout but the value of variable changes. Equations can be solved with the help of various methods. The method of elimination can also be used.

It is very simple to use and can be easily learnt. One of the variables is eliminated to find the value of the other variables in this method. The equation algebra can be an interesting subject to learn if one has interest to solve equations. Sometimes equations in algebra can be very tricky to solve and one has to be careful while solving them.

The concept equations algebra is nothing but the equations that are present in this discipline of mathematics. There can be various methods to solve these equations. Many methods are available online to solve them. These methods can be easily learnt and applied while solving them. The algebra equations online will help in explaining the concept better.

Once the equations are solved, various values of the variables present in the equations are obtained. These values must satisfy the whole equation; otherwise they cannot be taken as the solution of the equation. These solutions must be feasible and must be acceptable.

An equation can have more than one solution. Once the solutions are obtained, it can be checked whether the solutions are right or not. Sometimes even complex numbers can be obtained as part of the solutions. In a complex number there is an imaginary part attached to it. There is also a real part also to the complex number. Both the real and imaginary part forms the complex number.

The art of solving equations can be very interesting. The equations can be of different types. There can be linear and quadratic equations. The difference between the two is only in their degree. The linear equation is very simple to solve compared to the quadratic equation.

The linear equation can be easily solved with the help of basic arithmetic operations and the principle of transposing. Once these concepts are known the linear equations can be easily solved. Once solved the solutions could be easily checked.

Boolean Algebra Tutorial

This Boolean Algebra Tutorial gives us a holistic view of what Boolean functions and calculations are and will surely aid in solving many mathematical problems related to Boolean algebra
Basic operations in Boolean algebra are as follows:
Conjunction: also denoted by an inverted V this operation basically is similar to the intersection operation. In this we have a specific formula. Actually this formula is derived by basic algebra intersection operation:
A intersection B = A + B – A V B - - - - - - -(1)
Here V is the union operation.
In Boolean Algebra we only deal with true or false here. So anything that is not true will be a false. We can also denote it by zero and one. Here one will denote a true and zero will denote a false.
Now let us state the complimentary operation. This is also called as the negation operation. It is a singular operand operation. That is in this we will need just one operand to calculate the result. Let us take an example. Say A = True. Now negation A will be false. This operation can be stated as :
Let A = True;
~ A
Output A. Now the value of A will be false.
Union operation has already been discussed. In Boolean we have certain Boolean Algebra Rules to be followed for the same. Let us take a quick look at them.
True union True = True
(True union False) or (false union true)= True
False union False = false

Similarly for intersection we have
True intersection true = true
True intersection false = false
False intersection false = false
As we can see both the operation are just opposite to each other. These are Boolean form of algebra rules. Let us state them in a more simplified way.
A + 1 = 1
A + 0 = A
A . 1 = A
A . 0 = 0
So it is all about the multiplication or the dot product and the addition with zeros and ones. A here can have any value that is one or zero. Boolean Algebra Examples can be:
Let a = 1
B = 0 so a . b = 1 . 0 = 0
Similarly let b = 0, ~ b = 1. This is an example of complimentary operation.
Other rules for the Boolean forms are:
~(~A ) = A
A + ~A = 1
A. ~A=0
~(A+B) = (~A) .(~B)
~(A.B) = (~A) + (~B)

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