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. He published it 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. In digital electronics there are several methods of simplifying the design of logic circuits. This algebra is one of these methods. 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.
According to Cumulative Law, the order of OR operations and AND operations conducted on the variables makes no differences.
Let us show one use of this law to prove the expression
Proof:
From truth table,
From truth table it is proved that,
Proof from truth table,
Both A and A+A.B column is same.
Proof from truth table,
Both A and A.X or A(A+B) column are same.
De Morgan’s Therem,
Proof from truth table,
These are another method of simplifying complex Boolean expression. In this method we only use three simple steps.
will be first complemented, i.e.
.Now, change all (+) to (.) and (.) to (+) i.e.
Now, complement each of the variable,
This is the final simplified form of Boolean expression,And it is exactly equal to the results which have been come by applying De Morgan Theorem.
Another example,
By Second Method,
Representation of Boolean function in truth table.
Let us consider a Boolean function,
Now let us represent the function in truth table.
Thus we have shown some basic laws of Boolean algebra. In the other page we have described De Morgan’s theorems and related laws on it.
There are also few theorems of Boolean algebra, that are needed to be noticed carefully because these make calculation fastest and easier. Boolean logic deals with only two variables, 1 and 0 by which all the mathematical operations are to be performed.
Boolean algebra or switching algebra is a system of mathematical logic to perform different mathematical operations in binary system. There only three basis binary operations, AND, OR and NOT by which all simple as well as complex binary mathematical operations are to be done. There are many rules in Boolean algebra by which those mathematical operations are done. In Boolean algebra, the variables are represented by English Capital Letter like A, B, C etc and the value of each variable can be either 1 or 0, nothing else. In Boolean algebra an expression given can also be converted into a logic diagram using different logic gates like AND gate, OR gate and NOT gate, NOR gates, NAND gates, XOR gates, XNOR gates etc.
Boolean algebra or switching algebra is a system of mathematical logic to perform different mathematical operations in binary system. There only three basis binary operations, AND, OR and NOT by which all simple as well as complex binary mathematical operations are to be done. There are many rules in Boolean algebra by which those mathematical operations are done. In Boolean algebra, the variables are represented by English Capital Letter like A, B, C etc and the value of each variable can be either 1 or 0, nothing else. In Boolean algebra an expression given can also be converted into a logic diagram using different logic gates like AND gate, OR gate and NOT gate, NOR gates, NAND gates, XOR gates, XNOR gates etc.
Some basic logical Boolean operations, AND Operation
OR Operation
Not Operation
Some basic laws for Boolean Algebra
A . 0 = 0 where A can be either 0 or 1.
A . 1 = A where A can be either 0 or 1.
A . A = A where A can be either 0 or 1.
A . Ā = 0 where A can be either 0 or 1.
A + 0 = A where A can be either 0 or 1.
A + 1 = 1 where A can be either 0 or 1.
A + Ā = 1
A + A = A
A + B = B + A where A and B can be either 0 or 1.
A . B = B . A where A and B can be either 0 or 1.
The laws of Boolean algebra are also true for more than two variables like,
OR OperationNot OperationSome basic laws for Boolean AlgebraA . 0 = 0 where A can be either 0 or 1.A . 1 = A where A can be either 0 or 1.
A . A = A where A can be either 0 or 1.
A . Ā = 0 where A can be either 0 or 1.
A + 0 = A where A can be either 0 or 1.
A + 1 = 1 where A can be either 0 or 1.
A + Ā = 1
A + A = A
A + B = B + A where A and B can be either 0 or 1.
A . B = B . A where A and B can be either 0 or 1.
The laws of Boolean algebra are also true for more than two variables like,
Cumulative Law for Boolean Algebra
According to Cumulative Law, the order of OR operations and AND operations conducted on the variables makes no differences.Associative Laws for Boolean Algebra
This law 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 operators.Distributive Laws for Boolean Algebra
This law is composed of two operators, AND and OR.Let us show one use of this law to prove the expressionProof:Redundant Literal Rule
From truth table,| Inputs | Output | ||
| A | B | ĀB | A + ĀB |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 |
| Inputs | Output | |
| A | B | A+B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Absorption Laws for Boolean Algebra
Proof from truth table,| Inputs | Output | ||
| A | B | AB | A+A.B |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
Proof from truth table,| A | B | A+B | A.X(A+B) |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 |
De Morgan’s Therem,Proof from truth table,Examples of Boolean Algebra
These are another method of simplifying complex Boolean expression. In this method we only use three simple steps.- Complement entire Boolean expression.
- Change all ORs to ANDs and all ANDs to ORs.
- Now, complement each of the variable and get final expression.
will be first complemented, i.e..Now, change all (+) to (.) and (.) to (+) i.e.Now, complement each of the variable,This is the final simplified form of Boolean expression,And it is exactly equal to the results which have been come by applying De Morgan Theorem.Another example,
By Second Method,Representation of Boolean function in truth table.Let us consider a Boolean function,
Now let us represent the function in truth table.Thus we have shown some basic laws of Boolean algebra. In the other page we have described De Morgan’s theorems and related laws on it.
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