Boolean Algebra Tutorial

This Boolean Algebra Tutorial gives us a holistic view of what Boolean functions and calculations are and will surely aid in solving many mathematical problems related to Boolean algebra
Basic operations in Boolean algebra are as follows:
Conjunction: also denoted by an inverted V this operation basically is similar to the intersection operation. In this we have a specific formula. Actually this formula is derived by basic algebra intersection operation:
A intersection B = A + B – A V B - - - - - - -(1)
Here V is the union operation.
In Boolean Algebra we only deal with true or false here. So anything that is not true will be a false. We can also denote it by zero and one. Here one will denote a true and zero will denote a false.
Now let us state the complimentary operation. This is also called as the negation operation. It is a singular operand operation. That is in this we will need just one operand to calculate the result. Let us take an example. Say A = True. Now negation A will be false. This operation can be stated as :
Let A = True;
~ A
Output A. Now the value of A will be false.
Union operation has already been discussed. In Boolean we have certain Boolean Algebra Rules to be followed for the same. Let us take a quick look at them.
True union True = True
(True union False) or (false union true)= True
False union False = false

Similarly for intersection we have
True intersection true = true
True intersection false = false
False intersection false = false
As we can see both the operation are just opposite to each other. These are Boolean form of algebra rules. Let us state them in a more simplified way.
A + 1 = 1
A + 0 = A
A . 1 = A
A . 0 = 0
So it is all about the multiplication or the dot product and the addition with zeros and ones. A here can have any value that is one or zero. Boolean Algebra Examples can be:
Let a = 1
B = 0 so a . b = 1 . 0 = 0
Similarly let b = 0, ~ b = 1. This is an example of complimentary operation.
Other rules for the Boolean forms are:
~(~A ) = A
A + ~A = 1
A. ~A=0
~(A+B) = (~A) .(~B)
~(A.B) = (~A) + (~B)

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Vector Components

As we know Vector is a quantity which has some magnitude and direction. Any Vector can be split into two Vector components such that the two Vector  component are perpendicular to each other.
For example as shown in the figure given below a Vector in the north west direction can be split into two components which are in north and west direction. Similarly a Vector in north east direction can be divided into two components which are in north and east direction. Note that the magnitude of these two components will not be same as that of the initial Vector.

Vector Components and the Vector which is divided into components together form a right angled triangle such that the two components are base and altitude of the triangle and original Vector if hypotenuse of the triangle. Given below a Vector ‘a’ which is divided into two components ax and ay such that ax, ay, a form the base height and hypotenuse of the triangle. ay component shown in figure can be shifted in right direction parallel to form the height of the triangle.  
Components of a Vector can be found out by using properties of trigonometry which are based on a right angled triangle.

According to trigonometry, in a right triangle if hypotenuse makes an angle Ө with the base then,
sinӨ = height or perpendicular/hypotenuse
cosӨ = base/hypotenuse.
For previous illustration sinӨ = ay/a
cosӨ = ax/a
So, Vector Component ax and ay will be:
ax = a(cosӨ)   ……….(component of Vector ‘a’ in x direction)
ay = a(sinӨ)  ………….(component of Vector ‘a’ in y direction)
Vector a can be written as: a = axi + ayj = a (cosӨ)i + a (sinӨ) j
If a Vector is given as a = a1i + a1j, then the Vector components can be read directly. Here a1 is component in x (horizontal) direction and a2 is Vector in y (vector) direction.
Let us take an example of Components of Vectors :
Example) Find the component of a Vector with magnitude 5 and makes an angle of positive 60o with horizontal.
Solution) let the Vector be P.
components of Vector P in x direction will be: Px= PcosӨ
= 5cos(60o)
= 5(1/2)
= 5/2 = 2.5
components of Vector P in y direction: Py= P(sin60o)= 5(√3/2)
So Vector P = 2.5i +5(√3/2)j
Note) If the Vector makes 60o with the vertical then Py = Pcos(60o) and Px = Psin(60o) or Py = Psin(30o) and Px=Pcos(30o) (This is because in this case the Vector will make 30o with horizontal)

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.