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.

Negations in Mathematics

Previously we have discussed about how to solve limits and In today's session we are going to discuss about Negation in math,it is used to represent the negative value of any number and comes under icse books. Word negations itself represents the opposite value of any given value and are used in almost every field of mathematics even in math word problems.
Let us understand it through some examples: If a statement is given " Rise in Temperature by 5 degrees" , then its Negation will be " Fall in the temperature by 5 degrees"
If the rise in the temperature is represented by + 5 , then the fall in temperature would be represented by -5.
Similarly, if we write " deposit Rs 200 in the account" , it is represented by + 200, then its negation would be " withdrawal of Rs 200 from the account"  and it will be represented by -200.
Another statement " 250 meters above the sea level " is represented by + 250 m  its negation will be " 250 meters below the sea level " is represented by - 250 m.
Application of negation is seen in each and every sphere of life. In every mathematical expression , we come across negation  where inverse of any value is taken care of.(want to Learn more about Negations ,click here),
 We also observe that the sum of positive and negative of any number is always zero. Thus +25 + (-25 ) = 0. There are many instances to see the application of negation for solving the problems related to integers. For example, let us consider the problem:
The temperature in Jaipur was observed 12 degrees at 9:00 am, which shot up by 10 degrees at 12:00 noon and then fall down by 15 degrees at 8:00 pm. what was the temperature at 8:00 pm?
To solve such problems we consider rise in temperature as a positive number and the fall in the temperature as a negative integer and then  proceed as follows
= 12 + (+10) + ( -15)
= 22 + (-15)
= 7 degrees Ans.
This is all about the Negations in Mathematics and if anyone want to know about Discreet Mathematics then they can refer to Internet and text books for understanding it more precisely. Read more maths topics of different grades such as Binomial Experiments in the next session here. 

How to Tackle Discrete Mathematics

Previously we have discussed about qualitative solution and In the study of discrete maths, we find that the knowledge of matrices is very important and it is the powerful tool which has variety of applications in our life and its an important part of CBSE math Syllabus. It just simplifies so many calculations.All problems related to solving linear equations can be easily solved by use of Matrices and determinants.
Now let us first see what is Matrix? Matrix is  a rectangular Array of m * n numbers in the form of m rows ( which are vertical lines ) and n columns, (which are  horizontal lines). The matrix is called m x n ( m by  n matrix ). We always enclose an array by [ ] or   (  ). The number of elements of the matrix are m*n. This is called the order of any particular Matrix. If we talk of more than one matrix, we say matrices. So word matrices is plural of Matrix.
3     5    7
2      3   6    is the matrix of 2 rows and 3 columns, so it is 2 x 3 matrix
To find the location of a particular element of the  matrix, we simply mention the row and column of which the element is taken. Lets say 3 is the ( 1, 2 )th element of the given matrix. Each element is represented as   a_{(ij) ,}    where I and j are the respective rows and columns.(Want to know more about discrete mathematics,Click here)  
Let us see an example. If there is a matrix of 12 elements, what are the possible orders of that matrix?
All possible orders of the matrix with 12 elements are ( 1, 12 ) , ( 2, 6 ), ( 3, 4 ), ( 4, 3 ), ( 6, 2 ), (12 , 1 ).
If we have to construct a 3 x 2 matrix, whose elements are given by
 a_ij = ( I + 2j ) .
Then , we first observe that the matrix is 3 x 2 , it means it has 2 rows and 3 columns. So the value of I = 1 , 2 , 3 and value of j = 1, 2 .
So a₁₁ = ( 1 + 2 * 1) = 1 + 2 = 3;   a₁₂ =( 1 + 2 *2 ) = 1 + 4 = 5
    a₂₁ =  ( 2 + 2 * 1 ) = 2 + 2 = 4 ; a₂₂ = ( 2 + 2 * 2) = 2 + 4 = 6
   a₃₁ = ( 3 + 2 * 1 ) = 3 + 2 = 5 ; a₃₂ = ( 3 + 2 *  2 ) = 3 + 4 = 7
so we get the matrix:  A =   3       4
                                            4       6
                                            5       7
We should always remember that a matrix is always represented by capital letter.

Scalar Matrix: A matrix where every non – diagonal element is zero and the diagonal elements are equal are called scalar matrix.

         5    0     0
         0     5    0           is the scalar matrix of order 3.
         0     0    5
 Unit Matrix: A matrix in which all non- diagonal elements  are zero and the diagonal element is “ 1” is called a unit matrix.
  1      0  is the unit matrix of order 2
  0      1
and
 1       0       0
0         1      0 the unit matrix of order 3.
 0        0       1
This is all about Discrete Mathematics and if anyone want to know about Compound Statements then they can refer to internet and text books for understanding it more precisely.Read more maths topics of different grades such as Congruence and Similarity in the next session here. 

Permutations in Discrete Mathematics

Hello friends,Previously we have discussed about what is calculus and today we are going to discuss a very interesting topic “discrete mathematics” which includes many topics of algebra like permutations, combinations, probability etc and falls under CBSE Board Syllabus. Discrete mathematics deals with the study of structures and curves which are not continuous or do not vary smoothly and is also very useful to solve math questions. For example study of sets is also included in discrete mathematics and we all know that a set is a well defined collection of objects which is nicely  constructed for a particular interval or condition. Such kind of finite studies are involved in discrete mathematics. permutations and combinations is the another topic included in discrete mathematics which also refers to the finite calculations. Friends, discrete mathematics covers many topics of mathematics. Here we are going to discuss in detail about a very important topic of discrete mathematics which is called permutations. Now let’s start with the definition of permutation. Friends, permutations refer to the different arrangements of a number of things. Basically study of permutations can be helpful to obtain the following calculations (want to Learn more about Permutations ,click here),
1.     To find the number of ways in which a task can be performed
2.     To find the number of words with or without meaning that can be formed by using some or all of the letters of alphabets
3.     To find the arrangement of ‘n’ things selected from ‘m’ different things
4.     To form the different numbers of different digits by using ‘n’ digits from a set of ‘m’ different digits
Friends calculation of permutations involves the method of factorial which is a very helpful and essential tool in the study of algebra. Factorial of a number n is written as n! and is defined by the following formula
N! = n.(n-1).(n-2).(n-3)……3.2.1
For more clear understanding of factorial let’s take an example to find the factorial of 5.
5! = 5.4.3.2.1 = 120 Friends, this is how the factorial of any number can be calculated. Now let’s have a look on factorials of few initial numbers which are generally used in the calculations of permutations. Students are advised to learn the value of these basic factorials directly
1! = 1.
2! = 2.1 = 2
3! = 3.2.1 = 6
4! = 4.3.2.1 = 24
5! = 5.4.3.2.1 = 120
6! = 6.5.4.3.2.1 = 720
Friends, in permutations the number of arrangement of r things taken from n things at a time can be calculated by an important formula ⁿp_r = n!/(n-r)!. Let's take few examples of permutations to understand this
Example: find the number of words that can be formed by 4 different letters of alphabets by taking 2 at a time without repetition of lattés.
This problem can be easily solved by using above formula as we have to form the words by taking 2 letters out of 4 we can find it by 4p₂ = 4!/(4-2)! = 4!/2! = (4.3.2.1)/(2.1) = 12. Now we can say that 12 words with or without meaning can be formed.

Friends this is how we can proceed to solve the problems of permutations and if anyone want to know about De Morgans law then they can refer to internet and text books for understanding it more precisely.Read more maths topics of different grades such as Binomial Property in the next session here.

Topics of Discrete Mathematics

Friends, for most of the students, the first and most common area of mathematics in college education is calculus. Calculus is the most important area of study of mathematics whose emergence causes the invention of modern mathematics and it was the key to most of the successful applications of mathematics in the science. It is basically a style of mathematics invented by Sir Isaac Newton. In general the words "calculus" means rock. Calculus is also very technical which takes a lot of time even to introduce its fundamental terms like continuity or derivatives.

Calculus is basically a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Basically it is the study of 'Rates of Change'. There are two main branches of calculus: Differential Calculus and Integral Calculus. Differential calculus determines the rate of change of a quantity and integral calculus finds the quantity where the rate of change is known.

Now I am going to give you a brief introduction of Discrete mathematics. It is a mathematics that deals with discrete objects. Discrete objects are those which are separated from each other. Integers, rational numbers that can be expressed as the quotient of two integers, automobiles, houses, people etc. are all discrete objects. On the other side the real numbers which include irrational as well as rational numbers are not discrete. As we all know that between any of the two distinct real numbers there is another real number different from either of them. So students can define them as they are equipped without any gaps and can't be separated from their immediate neighbors. In this way they are not discrete. Here in our Discrete mathematics syllabus I am going to discuss about objects such as integers, propositions, sets, relations and functions, which are all discrete along with the associated properties and relationships.

The important areas in applied mathematics comes with linear programming, coding theory, theory of computing are some of the applications that is collectively known as discrete mathematics. The most important thing to understand before going to solve a mathematics is that maths cannot be done without proofs.

Following are the topics to cover under Discrete Mathematics :

1. Boolean Functions and Computer Arithmetic
2. Logic
3. Number Theory and Cryptography
4. Sets and Functions
5. Equivalence and Order
6. Induction, Sequences and Series

Lets starts with the first topic:

1. Boolean Function and Computer Arithmetic: This section is further divided into two different sub parts that are :

Boolean Functions : In mathematics, it is a function of the form f : B^k →B, where B =0, 1 is a Boolean Domain and K is a non-negative integer which is called the arity of the function.

If K = 0 then the function is essentially a constant element of B.

It tells us how to find a Boolean value output based on some logical calculation from Boolean inputs. It plays an important role in questions of complexity theory as well as the design of circuits and chips for digital computers. Its properties play a critical role in cryptography, most commonly in the design of symmetric key algorithm. Following are the sub topics : Boolean function, binary operator, unary operator, not (∼), and (∧), or (∨), exclusive or (⊕), truth table, disjunctive normal form, conjunctive normal form.

Number Systems and Computer Arithmetic:

This sub topic covers following topics to cover digit symbols, digit symbol of index or rank i, base-b number, binary arithmetic, two’s complement, logic gate, half adder, full adder.

Now take on another topic that is:

2. Logic :

Reasoning conducted or compute according to strict principles of validity: which tells that "experience is a better guide to this than deductive logic". It is also defined as a particular system or allocation of the principles of proof and assumptions: it is also considered as "Aristotelian logic".

The sub topics of Logic part are:

Propositional Logic:

It is a formal system in which formulas of a formal language may be interpreted as representing propositions. Sentences considered in propositional logic are not approximate sentences, but the sentences or we can say them statement that are either true or false, but not both. If a proposition is true, then we say it has a truth value of "true"; if a proposition is false, its truth value is "false".

The topics to cover are:

Truth table that tells the statements or relations are true or false, statement forms, tautology, contradiction, implication, conditional, con- trapositive, double implication, bi-conditional, converse, inverse, if, only if, sufficient, necessary, unless.

Predicate Logic :

The propositional logic is not powerful enough to represent all types of assertions that are used in computer science and mathematics, or to express certain types of relationships between propositions such as equivalence. So we need Predicate logic , A predicate is basically a verb phrase arrangement that describes a property of objects, or a relationship among objects represented by the variables. Following are the sub topics which we are going to be learned in this section of mathematics: truth set, prime, composite, Fermat number, Mersenne number, perfect numbers, Goldbach conjecture, Fermat’s Last Theorem, Marin Mersenne (1588– 1648), Pierre de Fermat (1601–1665), Christian Goldbach (1690–1764), Leonhard Euler (1707–1783), Karl Friedrich Gauss (1777–1855) .

3 : Number Theory and Cryptography :

Number theory is a branch of pure mathematics which is basically formulated to the study of the integers. It consists of prime numbers (when multiplied give all the integers) as well as the properties of objects made out of integers or defined as abstractions of the integers. The sub topics are:

Basic Facts About Numbers :

Following are the topics which we are going to learn in this section of mathematics: rational numbers, irrational numbers, prime, composite, odd, even, n divides m, prime factorization, infinitely many primes, perfect squares, irrationality of integral square roots, residue classes mod d, mod as binary operator, mod as equivalence relation, modular arithmetic, modular addition, modular multiplication, floor function, ceiling function, diagonalization proofs

Cryptography and Secrecy:

These are some scientific techniques which are used too encrypt or decrypt the message to send forward. Following are the topics which we are going to learn in this section of mathematics :plaintext, ciphertext, key, espionage, greatest common divisor, least common multiple, greatest common division(m, n) as linear combination of m and n, Euclidean algorithm, Euler φ function, public key, symmetric encryption, discrete log problem, Diffie-Hellman algorithm, *RSA algorithm .

4 : Sets and Functions:

There are two sub topics that are:

Sets : It is a collection of well defined and distinct objects,

A set is a collection of well defined and distinct objects, acknowledged as an object in its own right. Sets are one of the most fundamental concepts in mathematics. It can be further defined as a collection of things that are brought together because they obey a certain rule. In this area of study we all are going to deal with following topics that are :intersection, union, difference, complement, symmetric difference, product, Cartesian product, binomial coefficients, C(n, k) = n , algebraic rules, associative rule, distributive rule, idempotent rule, DeMorgan’s rule, absorption rule, commutative rule, lexicographic order, power set, characteristic function, set partitions, Bell numbers, refinement

Functions:

Functions are basically mathematical ideas that hold one or more variables and produce a variable. In more appropriate manner we can say that a function associates one quantity, the argument of the function, also known as the input, with another quantity. The value of the function also known as the output. A function assigns exactly one output to each input. In this section we are going to learn about following topics that are:

domain, range, co-domain, image, relation, functional relation, one-line notation, surjection, onto, injection, one-to-one, bijection, permutation, two-line notation, composition of functions, cycle form of permutation, image of function, inverse image, co-image, set partitions, Stirling numbers S(n, k), Bell numbers .

5 : Equivalence and Order

In the equivalence section we study about equivalence relation, equivalence class, blocks of a partition, co-image, reflexive, symmetric, transitive, relational description, co-image description, pigeonhole principle, subset sums, monotone subsequences, extended pigeonhole principle, transpositions

In Order section we study about following areas of study:

antisymmetric relation, order relation, partially ordered set, incidence matrices, total ordering, linear ordering, set inclusion, lattice of subsets, incomparable subsets, refinement relation, direct product, Cartesian product, coordinate order, characteristic function, directed graph diagrams, transitive closure,

Boolean product, Boolean sum, covering relation, chain, least element.

6 : Induction, Sequences and Series

Induction is a method of mathematical proof which is used to establish that a given statement is true in all positive integers. Sequences is basically an ordered list of objects. Some of the topics to study in this section are induction hypothesis, induction step, base case, product of primes , infinite sequence, limit of sequence, convergent sequence, bounded sequence, monotone sequence, convergent to infinity etc.