Subsets

A collection of things which have something in common as per the rule is called a set. For instance set of colors would be {red, green, yellow, white, blue} this set represents the collection of five different colors, these are called the elements of the set.

The elements are separated using commas (,) and every element is unique in the set. The notation uses parenthesis which are curly or in other words the flower brackets {} and usually a set is denoted using a capital letter.  Let ‘E’ be a set denoting all the even numbers less than ten; so set E={2,4,6,8}. While learning about sets we come across finite set, infinite set, universal set, empty set, sub-set and power set.

In this article we shall learn more about a Sub-set. Consider a set A={1,2,3,4,5,6} and set B={2,4,6}. When we compare both the sets it is clear that all the elements of set B are present in the set A and hence we can call set B as the sub-set of set A.

Definition of a Subset can be given as, a set B is a sub-set of the set A only if every element of set B is in the set A. The subset sign is ‘⊆’.

For instance, set A={orange, pineapple, apple, grapes, kiwi, mango} and set B={apple, kiwi, orange, mango}. Here each element of set B is in the set A and hence Set B is a sub-set of Set A and is denoted as B⊆A. Suppose Set P={2,3,4,5,8} and set Q={1,3,7,9}; each of elements of set Q are not in the set P and hence we cannot call set Q a sub-set of set P.
So, subset meaning is the set which is a part of the whole set.

So, a subset in math can be better explained using the following example, a set P is a sub-set of set Q if and only if all the objects or elements of set P is in set Q.

If set P={x, y, z} and set Q={x, y, z, p, q, r}, every element of P is in Q and hence P⊆ Q. Some of the subsets examples are, list of all the sub-sets of the set A={a, b, c} can be given as { },{a},{b},{c}, {a,b},{a,c},{b,c}, {a,b,c}.

The number of sub-sets of a given set can be given by the formula 2^[n(S)] where [n(S)] is the number of elements in the set. If there are three elements, the number of sub-sets is 2^3 which is 8 sub-sets in all.

Interval notation

A set in mathematics can be defined as collection of elements. There are different ways in which a set of numbers with end points can be described, roster form, set builder form, interval notation, graphing on a number line and using venn diagrams. Interval notation is a method of representing a set of numbers which describe the span of numbers that lay along an axis namely the x-axis.

An interval can be defined as a subset of the numbers. There are two symbols used in interval notation, an open bracket or parenthesis denotes an open interval in which the number is not included and a square bracket or parenthesis denotes a closed interval in which the number is included.

For instance, (-3,3) is an open interval notation in which both the numbers -3 and 3 are not included in the span of numbers and a closed interval notation is given by [-3,3] where both the numbers -3 and 3 are included in the span of numbers.

For more interval notation practice let us consider the following examples, [-4,1)= {-4, -3,-2,    -1 ,0,1} , this is a half open interval. (-2,1] is a half open interval which is equivalent to set of numbers
 {-1,0,1}.
Let us now learn the steps in solving inequalities in interval notation given the inequality x>-4, here the inequality symbol is greater than but -4 is not included, so it is an open interval and also it is never ending which means it extends till infinity.
So, the interval notation would be (-4,-infinity). Let us now graph and write in interval notation the inequality x<=-2.

In the above graph the closed red dot over -2 shows a closed half interval notation which means -2 is also included in the interval and as the inequality is less than -2 it is towards the left on the number line which shows the other end point is never ending and hence is minus infinity.
Together the half interval notation of the given inequality would be (-infinity, -2]. If there is an open dot over the number on a number line it denotes the number is not included and hence the symbol ‘(‘ or ‘)’. So, inequalities in interval notation are very simple when used with the appropriate parenthesis according to the given inequality.
For instance, 0<=x<6 as="" be="" can="" shown="" x="">-1 would be (-1,+infinity). Absolute value interval notation of |x|< 8 is equivalent to any real number between -8 and 8, the double inequality would be -8

Even and Odd numbers

Even numbers and odd numbers are two of the most important concepts in basic mathematics taught in primary school learning. Even and odd numbers are two basic classifications of numbers. Let’s have a look at both the concept along with relevant examples in this post.
Even Numbers
Even numbers are those numbers that can be evenly divided into two parts. In simple terms, any number that is divided by 2 is an even number such as 2, 4, 6, 8, and 10 and so on. For example:
• After comparing the 2 Christmas gift ideas given by Maria and You, I found yours better. (Here, 2 Christmas gift ideas is an even number).
• I have listed 8 common hobbies for children for my next article. (Here, 8 common hobbies for children is an even number).
• Are you turning 42 this year? (Here, 42 is an even number).
• I have bought 4 kgs of apples. (Here, 4 kgs is an even number).
• She bought the shoes for Rs.350. (Here, Rs.350 is an even number).
Odd Numbers
Odd numbers are those numbers that cannot be evenly divided into two parts. In simple terms, any number that is no divided by 2 is an odd number such as 3, 5, 7, 9, and 11 and so on. For example:
• He bought 3 gifts for his baby last week. (Here, 3 gifts is an odd number).
• The flight ticket’s range to my native is currently Rs.6789 (Here, 6789 is an odd number).
• I have bought 3 pairs of Yonex badminton shoes for my kid and his friends. (Here, 3 pairs of Yonex badminton shoes

is an odd number).• She is turning 7 years old this year. (Here, 7 years old is an odd number).
• Maria’s weight was 57 kg recently. (Here, 57 kg is an odd number)These are the basics about even and odd numbers.

Matrix: Cofactors

Consider a matrix that is a square matrx. That means that the matrx is n X n matrx. The number of rows and the number of columns in such a matrx are equal. Each term of the matrx can then have a cofactor. The position of each of the element of a matrx is described by (i,j). Where ‘i’ is row number (ith row) and j is the column number (jth column). For example consider the matrx below:
[a b c]
[d e f]
[g h l]

Here the position of the element a is (1,1) since it is the first element in the first row and first in the first column. Similarly the position of the element f would be (2,3) since it is the third element in the second row. So row number of f is 2 and column number is 3.

Cofactor of matrix:
To be able to find the cofactors of a matrix, we first need to find the minors of each of the elements. This can be done as follows:
If we take any entry, say ‘a’ in the above matrx and remove the row and the column containing a and keep the other entries in the same order, we get the determinant
|e f|
|h l|

This determinant is called a minor of ‘a’. Thus, removing the column and the row containing a given element of a matrx and keeping the surviving entries as they are, yields a determinant called the minor of the given element.

If we now multiply the minor of the entry in the ith row and jth column by (-1)^(i+j), we get the co-factor of that element. Therefore in the above matrx, the minor of h is
|a c|
|d f|
And multiplying this by (-1)^(3+2) = (-1)^5 = -1, we get the cofactor of the element h. This cofactor, therefore is
(-1) * |a c| = -(af – cd)
           |d f|

Note: The co-factor of an element is precisely the factor by which that element is multiplied in the expansion of the determinant of that matrx.

The co factors of a,b,c,… etc are denoted by Ac, Bc, Cc,… and so on.

If every element in a matrx is replaced by its cofactor, the resulting matrx is called the cofactors matrix. Therefore for our matrx above, the cofactors matrx would look like this:
[Ac Bc Cc]
[Dc Ec Fc]
[Gc Hc Lc]

Where, Ac = +(el-fh), Bc = -(dl-fg), Cc = (dh-eg), Dc = -(bl-ch) and so on.

Solving by elimination method

The elimination method is one of prominent methods in solution to the system of linear equations with two variables. The elimination method for solving equations may be apparent in certain cases where, the coefficients of a particular variable or variables are same.

In such a case you automatically use this technique. These cases of equations are called solving elimination equations. For, example, if x + y = 5 and x – y = 1, you eliminate ‘y’ by just adding both the equations to get the solution as x = 3. Similarly you eliminate ‘x’ by subtracting the second equation from first and figure out y = 2. But all systems are not as simple as this. Hence let us take a broader look on solving by elimination method.

Suppose, you have a system of equations in which all the coefficients of the variables are different. Select a variable for elimination. Your skill and experience will tell you which will be the ideal variable to eliminate. Take the LCM of the coefficients of that variable and multiply each equation with the ‘missing’ factor of the LCM in each case.

Now the given set of equations is transformed in such a way that the new set has the same coefficient for the selected variable in each equation. Now do the subtraction or addition operations on the transformed equations, so that the selected variable is eliminated and you get the solution of the remaining variable. Subsequently, plug in that in any of the given equations to figure out the solution of the other variable.

Let us elaborate with an example. Let the system be 17x + 2y = 49 and 19x + 3y = 54. It is prudent to decide to eliminate ‘y’. Eliminating ‘x’ not an incorrect step. But that will lead to cumbersome working. The LCM of 2 and 3 is 6.

Multiplying the first by 3 and the second by 2 (the missing factors of 6), the equations are transformed to 51x + 6y = 147 and 38x + 6y = 108.. A subtraction operation gives you the result as 13x = 39, means x = 3. Now plugging in x = 3 back, you can figure out y = -1.

The method of solving equations by elimination is the basis for the concept of use of determinants for solution of system of equations.

let us make a simple illustration. Suppose a1x + b1y = c1 and a2x + b2y = c2 . By elimination method you can establish, x = (c1b2 – c2b1 )/(a1b2 – a2b1 ), which is the concept behind Cramer’s rule of solution by using determinants.

Sequence and series

Sequence Series is a very important part of mathematics. When we write numbers in an order then we call it sequence of numbers.

The list of numbers follows a particular trend in that. The numbers in a seq are called terms of the seq. We cannot flip the terms in a seq.

Their order cannot be changed. We do the counting of terms from left to right.
For example: - 1, 3, 5, 7, 9 is a seq of five terms where first term is 1 and third term is 5 and last term is 7.

If we look at this seq we see that this seq follow a certain trend and that is every next term is two more than the previous term. Now what is a series?

A series is when we add the terms of a seq that constitutes to form a series. There are different types of seq and ser that exists and they are: -

1. Arithmetic Sequences and Series– In this type of seq and ser, the next term can be determined by adding the common difference to the previous term. The general arithmetic seq and ser is given as a, a+d, a+2d, a+3d, a+4d… where ‘a’ is the first term and d is the common difference. For example: - 1, 4, 7, 10, 13, 16. Here we see that first term is 1 and the common difference is 3

2. Geometric Sequences and Series– In this type of seq and ser, the next term can be determined by multiplying the common multiple by the previous term. The general geometric seq and ser is given as a, ar^2, ar^3, ar^4… where ‘a’ is the first term and r is the common multiple. For example: - 2, 6, 18, 54. Here we see that the first term is 2 and the common multiple is 3 which can be determined by dividing any term by its previous term.

3. Infinite Sequences and Series– In this type of seq and ser, the last term is unknown to us. We do not have fixed number of terms in that as it is an infinite ser. For example: - 2, 6, 10, 14…..

Series and Sequences Formulas are used to determine the first term, last term, sum of terms, number of terms or the common difference or ratio.

Equivalent Decimals

Decimals are nothing but a whole numbers but broken into pieces, so example we have one whole block, and we cut it in 2 pieces, and we call it as Tenths. How to do equivalent decimals, so how do we write it ? We write it as 1 .2 tenth (Here decimal represents AND). But if we break that down even further into more ten pieces and take out and add 3 then we call these as Hundredths.  So this way we have once, tenths and hundredths. So this way we will write it as 1.13 (one and thirteenth, remember here the last piece that we saw. Here we will name the last 1 and 3 as thirteenth, the last digit even if it is zero we call it that). Now late us take one more piece of cube and break it down into ten equal parts that will be called as thousands. So we have ones, tens, hundredths and thousands. Let us say we have this number up in broken to two pieces .we will call it as 1.232 that is one and two thirty two thousands, so we read the decimal like this way.

Now moving up to equivalent decimals examples, what if we have only one block so we will call it as 0.1( here we will read it as zero and one tenth.) but let us say we want to turn it to hundredths. The whole one piece when broken into ten equal parts has the same value for 0.1 as well. However they are hundredth. so all we need to say hundredth we need to add a zero. Because when we read decimals we need to say the last digit. Thus it will be 0.10(we call it as zero and one tenth one hundredth.) so one tenth is equal to tenth hundredths.

How to find equivalent decimals? We can by taking the denominator and dividing it with the numerator of the same fraction. Now for example we have ½ as a fraction so, 2 have to get divided by 1 so we will add a zero and bring up a decimal point. This way we have 10 which will be divisible by 10. As 2 times 5 is equals to 10.  And 10 subtracted by 10 and remainder comes to zero. So we have got the solution as 0.5 is the equivalent decimal.
What is a Equivalent decimal are nothing but the decimals which have the equal proportions.