Understanding the laws of Boolean Logic

There have been various laws in the arithmetic number system. The study of these laws aids in the solving of various mathematical problems. The Boolean algebra laws are the laws concerning the logic 0 and 1. Only zero and one are used in the Boolean algebra. Sometimes true or false is also used to represent the same logic. Ultimately they represent nothing but on and off of supply of flow of current in an electric circuit.

The operations used in this algebra are similar to the laws in the algebra. There are certain basic differences and the notations might change. But the process is similar and simple. The Boolean laws can be conjunction, disjunction and negation. Conjunction is nothing but the process of multiplication that is present in the real numbers algebra. Dis junction is the process similar to addition that is done in the real number algebra. Negation is nothing but taking the negative of the given value. If the given value is x, then it is replaced by ‘-x’ in the process of negation.

This algebra was developed by Boole in the year 1840 and that is why it is known as Boolean algebra. It is used in digital electronics. It is very helpful in the analysis of various gates and circuits. It has certain laws and these have to be learnt to understand and solve the problems. The Boolean laws are to be learnt for this and they are quite simple to learn. There can be different variables used in this algebra but these variables can have only two values zero and one. A large expression can be formed with the help of the variables.

These are to be solved with the help of laws of Boolean algebra and understood. The variables like A, B, X, Y can be used to represent the variables in this algebra and then the expressions formed and the logical operations carried out. Any variable added to ‘1’ gives one. This is one of the laws. Any variable added to zero gives the same variable. Any variable multiplied with ‘1’ gives the same variable. The same variable if multiplied with ‘0’ gives ‘0’. In electronics it represents an open circuit. Two similar variables on addition give the same variable. Multiplication of two similar variables gives the same variable. Commutative property of addition also holds good. These are some of the laws. The variables can take values 0 and 1 only.

Department of education

The department of education in math gives us various contents and syllabus. The math can be differentiated into various topics along the grade of the class. The department of education shows the step by step development in grade wise for the math. Algebra has differentiation when compared to the grade 5 and grade 12. The math can be classified into various topics as the education depends.

Department of Education:

Let us have the topic named partial product which is made to differentiated based on the grade. The lowest grade product gives the multiplication table they are,

1 x 2 = 2.

9 x 4 = 36.

As they have shown for the higher grade the multiplication can be made as the partial product math. These give the suitable example for the department of education of math.

The partial products math is nothing but the summing of the two terms not only through the addition but also through the product method. First the terms are rounded and made to multiply with the left most term of the number. Then the second term is made to rounded and the values are made to multiplied with the another term of the left. Then the left most term can be made multiplied with left most term of another term. Then the right most term can be made to multiplied with the right most term of the another term.

At last the values are made to summing up and the total of the values gives the product of the both terms. This is how the product partial terms are executed. The partial math includes all the operations like addition and subtraction in the same manner. Then the second term is made to rounded and the values are made to multiplied with the another term of the left. Then the left most term can be made multiplied with left most term of another term.

Example Problems for Department of Education:

Example 1: Find the partial product of the term 93 x 25?

93

25

--------

90*20 -> 1800

90*  5 ->   450


3*20 ->     60

3*  5 ->     15

---------

2325

Answer: 2325

Writing and simplifying algebraic expressions

What is an algebraic expression?
A symbol in algebra that is supposed to have a fixed value is called a constant, where as any other symbol in algebra that can be assigned different values are called a variable. For example 1/3, -8, pi, e etc are all constants and x, y, z, etc are all variables.
A sensible combination of constants and variables conjoined by arithmetic signs of +, -, * and / is called an algebraic expression. The parts or terms of an algebraic expn are separated by + or – sign. The constants and variables that are connected by * or / signs are deemed as one term. For example, in the algebraic expression 2x + 3xy + 5y - 7 there are four terms. 2x, 3xy, 5y and -7.

Simplifying an algebraic expression:
Simplification of an algebraic expression involved addition, subtraction, multiplication and division. So, How to Simplify an Expression? The rule of PEMDAS that we follow for simplifying arithmetic calculations is also followed in simplifying algebraic expn as well. Like terms can be added or subtracted together. Like terms are the terms that have the same set of variables with same exponents.  For example, 2x, 5x, 0.75x etc are all like terms. They can be added or subtracted and combined to one single term. Whereas, 2y, 5x, 0.75z^2, x^2 etc are all unlike terms. They cannot be added or subtracted. If an algebraic expn has brackets, then they have to be simplified first. There is no clear cut step wise process to simplify algebraic expn as there can be innumerable different types of expressions, and each can be simplified in simple ways. Let us look at some examples to understand better.

Example 1: 
Simplify: 3x + 4y – 3 + 4x + 7y + 8
Solution: 
For this type of expression it is possible to simplify by combining like terms.
Step 1: collect the like terms together
=> 3x + 4x + 4y + 7y + 8 – 3
Step 2: now combine the like terms
=> (3x+4x) + (4y +7y) + (8-3) = 7x + 11y + 5
That is your answer.

Writing an algebraic expression:
For any given situation described in words, that involves numbers, we can write the corresponding algebraic expn.
Example: 
Write an expression for “twice a number added to 1”.
Solution:
Here, suppose the number is x. Then twice the number would be 2x and that when added to one gives us 2x+1. This would be our required algebraic expn.

Parabola Equation

A parabola is a conical section formed by the intersection of a conical right circular surface and plane which is parallel to the straight line generated at that surface. The parabola can also be generated by examining the point called as focus and the line called as directrix. The locus of all points in the plane which are at equal distances from both the point and the line is said to be as the parabola. The line which passes through the focus (line which splits parabola in the middle) and lies perpendicular to the directrix is known as the “Axis of Symmetry”. Also the point present on this symmetry of axis which will intersect the parabola is known as the “Vertex”. In this vertex point, the curvature will be always greatest. A parabola can be opened up and down, left and right or even in some other arbitrary directions. They can be rescaled or repositioned in order to fit exactly with any other parabola, which implies that all parabolas are similar.

Standard form of a Parabola Equation
The general form for finding the Equation of a parabola is given as,
Y = ax2 + bx + c, where ‘x’ and ‘y’ are the points on the parabola.
In the above equation, when the value of ‘a’ is greater than zero, then the parabola will open upwards and when the value of ‘a’ is lesser than zero, then the parabola will open downwards. Also, the axis of symmetry will be the line of ‘x’ value equaling to negative of ‘b’ divided by 2a.

Parabola Equation Vertex
The point where the parabola will cross its axis is simply said as the vertex of a parabola equation. From the above standard equation of parabola, it implies that when the coefficient of x2 term occurs as positive, then the vertex will be at lowest point drawn on the graph. Similarly, when comes with negative coefficient of x2, it will be at highest point which can be said to be at the “U’ shape top.
The vertex form of a parabola equation can be written as,
Y = a(x-h) 2 + k, where ‘h’ and ‘k’ are the vertices of a parabola.

The parabola can also form into horizontal direction in the graph extending through the horizontal axis. The horizontal parabola Equation is simply as same as the standard form of parabola equation.

There are numerous websites which provide Parabola Equation solver in which when we give the vertex and focus of the parabola, it will automatically generate the standard and vertex form of the parabola.

Concept of dependent variables in math and statistics

A dependent variable in math is a variable the value of which depends on one or more other variables. For example if we have an equation that looks like: y = 2x+ 3. Here, y is one such variable because the value of y would depend on what value is assigned to x. Such an equation is called an equation in two variables. When plotting such a relationship on a graph, the independent of the variable x is usually plotted on the x axis and the dependent-variable axis is usually the y axis. Therefore, if the relationship is like this: p = 3q + 7, then the independent of the variable q would be plotted on the x axis and the dependent of these which is p would be plotted on the y axis.

Dependent variable in an experiment can be compared to the output of the experiment. The independent of these variables is usually the input variable in any dependent random variables experiment. This definition of the dependent type of variable is by and large common throughout the world. However its application would vary a little depending on whether the experiment is statistical or is it just mathematics.
Dependent variables examples:

A medical research laboratory is studying the effect of a specific drug in treatment of cancer. Here the quantity of drug administered would be the independent-variable, and the affect the drug has on cancer would be the dependent of the variables.  This is also a statistical example of such dependent pattern variable.

The equation we saw above: y = 2x + 3 is a mathematical example. Here y is the dependent and x is the independent one. When talking of these dependent of the variables, another concept that needs to be considered is that of limited dependent variables and unlimited dependent ones.  This concept is more applicable to statistical models. There are experiments where in one independent of the variable affects more than one dependent patterns. These multiple dependent or responding variables may be limited to 2 or 4 or 10 or may be unlimited. For example if the amount of chlorine in a water supply system of a town is the responding variable, and we know that change in the chlorine amount would affect the people drinking that water, people using that water for washing clothes or utensils, the effect of such water on animals, plants, metal pipes that carry that water, etc. Therefore there are many dependent or responding variables.

Introduction of Coins as Money

Money is the only factor which every single works for and in this materialistic world one can buy anything and everything with money. Money can be termed as an essential factor for living in today’s world. There can never be a question What Coins are Worth Money since even a penny makes something complete. There are people in this world who seek for the same coin which many take it for granted and leave behind. To understand the worth of a penny one should know What is Money and the answer for that will explain and serve the purpose.

Coins are of different value according the number valued on it. Money Coins are of different size and of different value. Today, these coins are valued more, that is their value is increased earlier there has been just a low value coin and now the government has started providing coins of high value. The value of a coin can be well explained by a beggar who begs for a penny to get his / her breakfast. A coin lost will make the person understand the worth and value of it.

Coins are actually better for blind people, since it is made of steel and can be felt and sense the value of the coin just by touching it. Money Coins Pictures actually help in serving the above usage to avoid duplicate coins when given to a blind person. The best part is, a blind person can find out the money note the difference between duplicate and original.

The above is the advantage of coins, but there are disadvantages in using the coins. The coins are very small in size and it is possible for a person to lose the same out of negligence. And they make the wallet filled up and often spoil the wallet when piled up and not used. There are people who think that it is cheap if they use a coin to buy things in their daily life. The importance of coin is realized in bus when the ticket conductor has no change and we are forced to get down of the bus. The irritation increases when the same bus conductor takes some coins worth .50 or 1 rupee since he has no change to provide the passenger. The coin is good and very important to a section of people and another part take it for granted.

Conditional and Biconditional statements

A conditional statement is a statement which is performed by if true or false. For eg: if p and q are two propositions, "if p then q" is known as conditional statement or implication. A statement is called biconditional when it is expresses the idea that the presence of some property is a necessary and sufficient condition for the presence of some other property.

Conditional and Biconditional Statements:

Conditional statements and biconditional statements of different propositions may be obtained by conjunction, disjunction and negation of propositions.

Conjunction Statement:
If p and q are two propositions, then compound proposition, "p and q" is known as conjunction of the proposition. It is indicated by p q. The conjunction of two propositions p and q are true, if both p and q are true and in all other cases it is false.

Disjunction Statement:
If p and q are two propositions, then the compound proposition "p or q" is called the disjunction of p and q. It is indicated by p v q. The disjunction p v q of two proposition p and q are false if both p and q are false and in all other cases, it is true.

Negation Statement:
Let p be any proposition. The suggestion "not p" is called the negation of p. It is indicated by ~p. The negation p is false if p is true and also the negation p is true if p is false.

Example for Conditional and Biconditional Statements:

Consider the proposition, “If it is rainy then it is cloudy”, which we say is a conditional statement.
Let us consider, p =“It is raining”, q =“It is cloudy”. Then the proposition can be written as “If p then q”. We symbolize this as, p à q. This can also be deliberated as “p implies q”. We never want something false to follow from something true; i.e. we do not want “If pq” to be true if p is not true and then q is not true.