Boolean Algebra Theorems and Laws of Boolean Algebra

Boolean algebra is a different kind of algebra or rather can be said a new kind of algebra which was invented by world famous mathematician George Boole in the year of 1854. In digital electronics there are several methods of simplifying the design of logic circuits. This algebra is one of the method which it can also be called is switching algebra. According to George Boole symbols can be used to represent the structure of logical thoughts. This type of algebra deals with the rules or laws, which are known as laws of Boolean algebra by which the logical operations are carried out. There are also few theorems of Boolean algebra, that are needed to be noticed carefully because it makes calculation fastest and easier. In Boolean algebra a digital circuit is represented by a set of input and output signals and the function of the circuit is suppressed as a set of Boolean relationship between the symbols. Boolean logic deals with only two variables, 1 for 'True' and 0 for 'false'.

We are familiar with the term Algebra which is related with general mathematics, but in this article we are dealing with Boolean algebra which is very much different from the original term. As in algebra we are accustomed with the general rules and formulas of basic addition, subtraction, multiplication and division but in case of Boolean algebra the implementation of abstract algebra and mathematical logic are more important. This algebra helps to define the logical operations of digital circuits. So it is a very important function of digital electronics and it helps in smooth running of the system.

History of Boolean Algebra

Now before discussing about the basics of Boolean algebra we should know about its history, who invented it, from where the original idea came. World famous mathematician George Boole invented Boolean algebra in the year 1854 in his book “An Investigation of the Laws of Thought”. Later using this technique Claude Shannon introduced a new type of algebra which is termed as Switching Algebra. There are specific rules and laws of Boolean algebra which are discussed below.

Before going to the laws of Boolean algebra and theorems of Boolean algebra we must know that know that Boolean expression can be stated as :-

i. A constant is a Boolean expression.

ii. A variable is a Boolean expression.

For example is B is a Boolean expression then B is also a Boolean expression. Variables such as BC and BCA + C are also Boolean expressions but A – B is not a Boolean expressions.

This type of expressions is formed by using binary constants, binary variables and basic logical operation symbols. In Boolean algebra an expression given can also be converted into a logic diagram using logical AND gate, OR gate & NOT gate. Basic logical operations are nothing but operations that include AND function i.e. logical multiplication, OR function i.e. logical addition and NOT function i.e. logical inversion.

Laws of Boolean Algebra
There are several laws in Boolean algebra.

As we know that this kind of algebra is a mathematical system consisting of a set of two or more district elements. There are also two binary operators denoted by the symbol bar (-) or prime (‘). These altogether follows the following laws:-

i. Commutative law.
ii. Associative law.
iii. Distributive law.
iv. Absorption laws.
v. Consensus laws.

i) Commutative law :- Using OR operation the Law is given as -

A + B = B + A

By this Law order of the OR operations conducted on the variables makes no differences.

This law using AND operation is -

A.B = B.A

This mean the same as previous the only difference is here the operator is (.). Here the order of the AND operations conducted on the variables makes no difference. This is an important law in Boolean algebra.

ii) Associative law : This law is given as -

A+(B+C) = (A+B)+C

This is for several variables, where the OR operation of the variables result is same though the grouping of the variables. This law is quite same in case of AND operator. It is

A.(B.C) = (A.B).C

Thus according to this law grouping of Boolean expressions do not make any difference during the AND operation of several variables. Though but these laws are also very important.

iii) Distributive law :- Among the laws of Boolean algebra this law is very famous and important too. This law is composed of two operators. AND and OR. The law is

A + BC = (A + B)(A + C)

Here the logic is, AND operation of several variables and then the OR operation of the result with a single variable is equivalent to the AND of the OR of single variable to one of the variable of several variables to make it simple, set BC be the several variables then A will be OR with B. Firstly and again A will be OR with C, then the result of the OR operation will be AND. The proof of this law in Boolean algebra is given below :-

Proof
A + BC = A.1 + BC [ Since, A.1 = A]

= A(1 + B) + BC [Since, B+1 = 1]

= A.1 + AB + BC

= A.(1 + C) + AB + BC [Since, A.A = A.1 = A]

= A (A + C) + B (A+C)

A+BC = (A+B)(A+C)

This law can also be for Boolean multiplication.

Such as - A.(B + C) = A.B + A.C

iv) Absorption laws : - In laws of Boolean algebra there are several groups of laws. Absorption laws are such a group of laws. The laws and their respective proves are given below.


i) A+AB = A

Proof. A+AB = A.1 + AB [A.1 = A]

= A(1+B) [Since, 1 + B = 1]

= A.1 = A

ii) A(A+B) = A

Proof. A(A+B) = A.A + A.B

= A+AB [Since, A.A = A]

= A(1+B)

= A.1

= A

iii) A+ĀB = A+B

Proof A+ĀB = (A+Ā) [Since, A+BC = (A+B)(A+C) using distributive law.]

= 1 (A+B) [Since, A+Ā = 1]

= A+B

iv) A.(Ā+B) = AB

Proof A.( Ā+B) = A. Ā+AB

= AB [‡AĀ = 0]

v) Consensus Laws :- This is other group of laws which are given more priority than the theorems of Boolean algebra. The Laws are as follows.

a) AB + ĀC+BC = AB+ĀC

AB+ĀC+BC = AB + ĀC + BC.1

= AB+ĀC+BC(A+Ā) [‡A+Ā=1]

= AB+ĀC+ABC+ĀBC

= AB(1+C)+ ĀC(1+B) [‡1+B=1=1+C]

= AB+ĀC

b) (A+B)( Ā+C)(B+C) = (A+B)( Ā+C)

Proof. (A+B)( Ā+C)(B+C) = (A+B)( Ā+C)(B+C+O)

= (A+B)( Ā+C)(B+C+AĀ) [ By distributive law]

= (A+B)(A+B+C)( Ā+C)( Ā+C+B)

= (A+B)( Ā+C) [‡A(A+B)= A]

The other small laws of Boolean are given below.

(a) A+0 = A (b) A+1 = 1 (c) A+A = A

(d) A+Ā = 1 (e) A.1 = A (f) A.0 = 0

(g) A.A = A (h) A. Ā = 0 (i) A' = A

Thus we have completed with the laws of Boolean algebra. In the other page we have describes on De Morgan’s theorems and related laws on it. Some problems given below will make your idea clear on this types of calculations.

Ex 1 :- Simplify the expression :-

Y = (Ā+B)(A+B)

Solution Y = (Ā+B)(A+C)

= C + A. Ā [‡(A+B)(A+C) = A+BC]

According to distributive law

= C+0

= C.

Therefore, Y = C

Here we have solved the equation using distributive law. We have to identify the equation first and then implement the suitable law for that particular equation. In the next step another law is used which is A. Ā = 0. Thus how we have solved it.

Simplify :-
Y = ĀBC+ĀBC+ABC+ABC

=ĀC(B+B)+AC(B+B)
= ĀC+AC

= C(Ā+A)

= C.1 = C

Thus Y = C

Here we have used very few laws of Boolean algebra but we have to identify the proper place and proper formulas while using those formulas. Look at the second line here (B+B = 1) this law is used. It is also used in the fourth line and thus the require solution of the given sum is found out in Boolean algebra.

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