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.