Distributive law and prepositions

Hello students, Previously we have discussed about subtracting integers worksheet and In today's session we are going to discuss about Distributive law and prepositions which comes under state board of maharashtra syllabus, In mathematics we study many laws like distributive, idempotent, complement, and associative laws. In this session, we are going to discuss the Distributive Law and Prepositions which are generally used in various fields of mathematics. These laws come in Boolean Algebra laws and are basic laws of the algebra too for showing logical equivalence (i.e. a kind of relationship between the two expressions).
The Common Distributive Law can be stated for all real number x, y and z as:
x ( y + z ) = x y + x z
The statements x ( y + z ) define the order in which we will add the y and z values then multiply x by the result.
The statements x y + x z define the order in which we multiply x and y, then multiply x and z and then add the multiplied results.(Know more about Distributive law in broad manner, here,)
Just take an example to understand it: -
x = 4, y = 5, z = 6
The distributive law is x ( y + z ) = x y + x z
Solution: - 4 (5 + 6 ) = 4 . 5 + 4 . 6
Note :- ( . ) dot operator specifies the multiplication.
4 ( 11 ) = 20 + 24
44 = 44
The Distributive law in terms of logical equivalence (explained with the help of prepositions which can be defined as symbols that show mathematical relationship between two statements) is:-
P ∧ ( Q v R ) = ( P ∧ Q ) v ( P ∧ R )
P v ( Q ∧ R ) = ( P v Q ) ∧ ( P v R )
Where preposition ∧ is used for AND, and v for OR operation.
So these laws describes the two different ways but both ways produce same results.
In the next session we are going to discuss online tutors homework help
and if anyone want to know about Congruence and Similarity then they can refer  Internet and text books for understanding it more precisely.

De Morgans law

Previously we have discussed about multiplying polynomials worksheet and In today's session we are going to discuss about Law of De Morgans which is a part of maharashtra higher secondary board syllabus, It interchanges an equation form to its negation form. Here negation means its opposite form. The law provides some rules that are known as transformation rules because it transforms an equation form to its negation form. By De Morgans law, we can relate conjuction and disjunction in terms of each other.
In simple english, Applying negation on the conjuction form of an equation gives us the disjunction of its negations. In other words, Applying negation on a disjunction form of an equation gives us the conjuction of its negations.

In logical language form,De Morgans law can be expressed as,
( A á´§ B ) ' = ( A ) ' á´  ( B )'
( A á´  B ) ' = ( A ) ' á´§ ( B )'

here ' stands for the negation of the expression.
á´§ stands for the conjuction of two expressions.
á´  stands for disjunction of two expressions.
= sign stands for equality,i.e we can replace one form to another form
The law has many forms.i.e, we can define the law in many ways.

This law has the same meaning in set (Data Set Example) theory.

The substitution form of this law is as follows ,

X . Y = ( X' + Y ' )'
X + Y = ( X' . Y ')'
. stands for conjuction and + stand for Disjunction .
' stand for the negation .

This law also holds for the functional forms as,
if we have two functional operator X and Y, then

X AND Y = NOT ( ( NOT X ) OR ( NOT B )
X OR B = NOT ( ( NOT X ) AND ( NOT Y ) )

AND operator stands for the conjuction and OR operator stands for the disjunction .(Know more about De Morgans law in broad manner, here,)
NOT operator stands for the negation.
In the next session we are going to discuss Distributive law and prepositions
and Read more maths topics of different grades such as Geometric Shape Attributes in the upcoming sessions here.

Boolean Laws

Hi friends, Previously we have discussed about slope worksheets and today we are going to learn about Boolean Laws, one of the most interesting and a bit complicated topic of mathematical world named as Boolean Algebra that comes into maharashtra education board. To solve Boolean Algebra problems what we need to do is hard work and a daily little practice. Boolean laws and theorems are also needed to understand before starting to solve boolean algebra.
Boolean Algebra :-It was developed by George Boole in year 1840 , it represents logical calculus of truth values and algebra of real numbers with operations like ab , a+b ,
Boolean algebra (or Boolean logic) is a logical calculus of truth value, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations, (Know more about Boolean Laws in broad manner, here,)
It is algebra of two algebra values where one can be either true or false
Boolean Laws:

Identity for Single Variable
Operations with 1and 0: 1. A+ 0 = A (identity) 3. A + 1 = 1 (null element)	2. A.1 = A 4. A.0 = 0
Idempotency theorem: 5. Y + Y = Y	6. Y.Y = Y
Complementarity: 7. X + X’ = 1	8. X.X’ = 0
Involution theorem: 9. (Y’)’ = Y
Identities for multiple variables
Commutative law:  10. A + B = B + A	11. A.B = B.A
Associative law: 12. (X + Y) + Z = X + (Y + Z)      = X + Y + Z	13. (XY)Z = X(YZ)        = XYZ
Distributive law: 14. a(b + c) = ab + ac	15. a + (bc) = (a + b)(a + c) Example :  (A+B)(AC+AC)+AB+B Solution :(A+B)A(C+C)+AB+B (A+B)A+AB+B A((A+B)+B)+B A(A+B)+B AA+AB+B A+(A+T)B A+B

Conjunction disjunction and Complement are the three basic operations which we can perform on Boolean values.
In The Next Topic We Are Going To Discuss Boolean Laws and if anyone want to know about Constructing geometry shapes then they can refer to Internet and text books for understanding it more precisely.

Bi Conditional Statements

Previously we have discussed about number line worksheets and In today's session we are going to discuss about Bi conditional statements which is a part of maharashtra state board of secondary and higher secondary education ac provides you excellent college algebra help. They are used to write the definitions in the geometry. In geometry definition is defined as statements for defining the mathematical objects and these statements are written in the form of bi conditional statements. For creating the bi conditional statements we combine the conditional statements and also converse it .
Bi Conditional statements are the statements that are written in the form of “ x if and only if y; “ that have the meaning that if x is true then y is also true. Bi conditional statements “ x if and only if y “ is also write as “ x iff y “ or x < - > y .
When both the conditional statement and its converse are true then its bi conditional statement is also true . When both the conditional and its converse are false then the bi conditional statement is also false .
Bi conditional statement examples:
If there are two statements x : A polygon is a triangle .
Y : A polygon has absolutely 3 sides .
So the problem based on these statements is determine the truth values of these statements :
( x → y ) > ( y → x ) .This compound statement is defined as the conjunction of the two conditional statements .In these type of statements the first statement is known as hypothesis and statement y is known as the conclusion .In the second statement y is the hypothesis and x is the conclusion .

The above statements are described by the truth table as :
x    y  x->y   y->x   ( x → y ) > ( y → x )
T    T    T          T        T
T    F    F          T        F
F    T    T          F        F
F    F    T          T        T
When both the statements have the same truth values then the compound statement
( x → y ) > ( y → x ) is also true . So , this is the way of combine the two conditional statements and have a bi conditional . In the next topic we are going to discuss Boolean Laws and if anyone want to know about How to solve Perpendicular equations then they can refer to Internet and text books for understanding it more precisely.

Compound Statements

Previously we have discussed about math equation solver and Today I am going to discuss about the compound statements in discreet math( you can also try compound interest calculator) which is a part of maharashtra secondary and higher secondary education board. Compound Statements in mathematics is very interesting and easy to learn. As we know that compound means combining two or more parts as to form a mix, so here we will learn to make the compound statements and how to solve compound statements. To compose a compound statement there are various ways. For this we have many connectors such AND, OR, IF….THEN, NOT etc.

Instead of using a simple statement again and again we use a single letter which represent whole thing. For example: (want to Learn more about Compound Statements, click here),

Statement: “I have money.”
A : “I have money”.
 We can replace this statement by ‘A’. So we don’t need to use this statement again and again, instead of this we simply use ‘A’.

To construct Compound statements we use following connectors:
1.     NOT: This is known as negation. Example: "I am not busy" we can replace it with:
2.     AND: This is called conjunction.
 Example: Here A : “I am going to market”.
                                   B: “I am going to park”.

                       A>B: “I am going to market AND park”.

3.     OR: This is called disjunction.

Example: Here A: I will go to the market.

                        B: Amit will go to the park.

                        AVB: I will go to the market OR amit will go to the park.

4.     IF…. THEN: This is called Conditional.
Example: Here A: I want to buy cloths.
                                    B:I will go to the market.

A->B: If I want to buy cloths then I will goto market.

Now to understand the basic concepts let us take an example:
Given: p is true, q is false and r is true.
Now to determine truth value of this compound statement (
q ) ->r.

P	q	r		q	( q)->r
T	F	T	F	F	T

We construct a truth table for the given statement to determine truth value.
So here I explained about the Compound Statements and how to solve compound statements, next I will tell you about Bi Conditional Statements and if anyone want to know about Steps of Solving equations with decimals then they can refer to Internet and text books for understanding it more precisely.

Conditional in Discreet Math

Today I am going to tell you about Conditional in Discreet math. In discreet math conditional statements are one of the types of the compound statements. We can define a conditional statement as: If x and y are two statement variables, the conditional of y by x can be given as “If x then y” or “x implies y”. It is denoted as “x->y”.  It will be false in case if x is true and y is false, else it will be true. In this x is called hypothesis or antecedent and y is called conclusion or consequent of conditional statement.

We can determine x->y by the following truth table:

x	y	x->y
t	t	t
t	f	f
f	t	t
f	f	t

In a conditional statement when the “if part” of the statement is false, the whole statement will be TRUE, without worrying about the conclusion (that can be either true or false). It does not depend on the conclusion part of the statement.
For example: if 1 is equal to 2 then 2 is equal to 3.
Here we will take one more example to understand it:
1.    x->

x	y		x->
T	T	F	F
T	F	T	T
F	T	F	T
F	F	T	T

Using truth tables we can easily solve these conditional statements.
Now I am going to tell you some interesting facts about conditional statements. To represent IF…Then as OR and negation of a conditional statement we can write it as:
1.    x->y as
2.    And the negation of “if x then y” is equal to “x and not y”, that is <(x->y) equivalent to x>
In the next section I will tell you about the compound statements in discreet math.

Conjunction in discreet maths

Today I am going to tell you about the Conjunction. Conjunction is a very interesting part of discrete math, it is essentially a logical function of binary that having a two place logical operator. This function is applied on two or more logical operands which results in either false/true or 0/1. Conjunction in discreet maths always returns true (1) value if both of its operands have true value else it will return false (0). It is represented as AND. Its logical sign is >.
Here we will take some examples to understand the main concept behind this:
1.       Let there are two operands say A, B where A = 0 and B = 0 then A>B = 0.
2.       Similarly if A = 1 and B = 1 then A>B = 1.
3.       If A = 0 and B = 1 then A>B = 0 and vice versa. (To get help on icse board books click here)

So from the above examples we get to know that whenever the value of A and B is 1(TRUE) then the result or conjunction of A and B is also TRUE (1) else FALSE (0).
Similar to Disjunction (OR) it also has some properties which help us to solve questions related to this. These properties are: Consider we have 3 logical operands say A, B, C then
1.       Commutative: Conjunction of A and B is equal to the conjunction of B and C i.e. A>B = B>A.
2.       Associative: According to this property A> (B>C) = (A>B) >C.
3.       Distributive: This property has two operations AND and OR. i.e. A>(BVC) = (A>B)V(A>C).
4.       Identity potency: According to this property Conjunction of an operator to itself always results in the same operator i.e. A>A = A.
So by using these properties of conjunctions we can solve any problems related to finding conjunctions. Today we got some interesting knowledge about conjunctions. In the next topic we will discuss about the disjunctions and In the next session we will discuss about Conditional in Discreet Math. 

Math Blog on Disjuction

Today I will tell you about this field of binary system: Disjunction in discreet math. Disjunction is very important and interesting term in discrete math. Discrete is a truth function of binary whose output will always be true if the applied input is also true, otherwise it will be false for input having false state. In logic and mathematics it is also known as the alteration, inclusive disjunction. We use OR to express it. A logical disjunction is basically is an operation that is applied on two logical values. These values always produce false value if and only if both the operands and values are false otherwise it is true if either one or both operands are true. This is a function having two literals which are separated by OR. Here are some examples of disjunction: (To get help on icse board syllabus click here)

If we have A = 0 and B = 0 then A V B = 0.
Similarly if we have A = 0 and B = 1 then A V B = 1.
Here are some properties of disjunction:
Let us take three operands which have some logical values true/ false or 0/1. These operands are A, B, C. So the properties of disjunction are followings:
1.       Commutative: According to this property the Disjunction of A and B is equal to the Disjunction of B and A i.e. (AVB) = (BVA).
2.       Associative: According to this AV(BVC) = (AVB)VC.
3.       Distributive: In this property of disjunction it uses two operations OR and AND i.e.  A V (B>C) = (AVB) > (AVC).
4.       Identity: In this property if we make OR of the same operand then we will get the same operand as the result. I.e. A V A = A
So from the above description we learnt about disjunction and some interesting properties of disjunction. In the next topic we will discuss about the conditional and In the next session we will discuss about Conjunction in discreet maths.

Discreet Mathematics

Hello, today we are going to discuss the discreet mathematics. Discreete mathematics is concerned with the study of distinct objects. Discreet mathematics provides the mathematical foundation for many courses in the field of Computer science, Operation research, Chemistry, Biology and other branches of Engineering. It involves the study of topics like Languages and logics, proof methods, graph theory, sets, relations and functions and permutations and combinations.

Let us have a look at some important terms of Discreet Mathematics.

MATRICES: A matrix is a rectangular arrangement of 'k' number of things. The entries in the rectangular arrangement (arrangement in rows and column) are enclosed in square brackets for example [a b c].
DETERMINANTS: For every square matrix there exists determinants which have the same elements as the matrix.The determinants of a square matrix, A= [a_ij]t can be written as |A|.
PERMUTATIONS: Permutations refer to the different arrangements of a number of things; Permutation is helpful to find the number of ways of doing a job. Number of ways or arrangements of n things taken r  things at a time can be found in this way:
       ⁿP_r= n!/(n-r)! Where n! = n. (n-1). (n-2)…………..3.2.1 
COMBINATIONS: Combination refers to the ways of selection of some or all of the objects from a group of objects, different groups of members, teams of players and effective selection of objects from n distinct objects can be easily calculated by using theory of combinations.
                                                    Solving discreet problems:
For practice purpose, here are few solved examples on permutations and combinations:

 Problem no.1: In how many ways can three prizes be distributed among 4 boys if

·         No one gets more than one prize.

·          A boy can get any number of prizes. 

Solution. there are total three prizes the first prize can be distributed in 4 ways and second and third prize can be distributed in 3 and 2 ways respectively as a boy can have only one prize so the total number of ways = 4.3.2= 24
 but when a buy can have any number of prizes every prize can be distributed in all 4 ways so total number of ways in this case are =4.4.4=64
Problem no.2. Given five different green dyes, four different blue dyes and three different red dyes how many combinations can be formed by selecting at least one green and one blue dye?
Solution. Number of ways of selecting at least one green dye
 =31
Now out of four blue dyes we can select  in
= 15, now after selecting one green and one blue dye we can select one or no red dye in
= 8, now the total no of ways = (31)(15)(8)= 372
In such a manner one can solve the various discreet mathematics problems .

Introduction to Basic Logical Operators

Previously we have discussed about law of exponential growth and In today's session we are going to discuss about mathematical logic concept which we will study icse books .Logic means reasoning, where reasoning may be legal opinion or mathematical confirmations. There are basically three BasicLogical Operators in mathematics - Negation , Conjunction and Disjunction and we will discuss their definition by the truth tables.

Negation (<) is a basic mathematical logic which is working like NOT operator means it gives opposite result of input like if input is yes then negation gives no as output.We can understand the logic of negation by truth table(want to Learn more about Logical Operators ,click here),

Input(x)      Output(< div=""> <>

True             False

False           True

We take some examples which are solved by negation

example 1 - problem : x is taller than y.

solution : x is not taller than y.

example 2 - problem : this is a leap year .

solution : this is not a leap year.

2. Conjunction(>) is a basic mathematical logic which is working like AND operation means if both conditions are true then it gives true else it gives false. Truth table of Conjunction between two inputs is

       Input    Input     Output(>)

          P           Q         (P AND Q)

       False   False       False

       True     False       False

       True     True        True

       False   True         False

3. Disjunction(V) is also a basic mathematical logic which is working like OR operation means if any condition is true is produces true as a result else in any other condition it produces false as a result and truth table of disjunction between two input are

      Input      Input     Output(>)

        P             Q          (P OR Q)

      False    False       False

      True      False       True

      True      True          True

      False    True          True

So, there are basically three logical operators which are commonly used in reasoning. So this is a brief discussion about logical operators. In the next article we will discuss about Negations in Mathematics and also about Frequency Distribution.