Boolean Algebra

In order to deal mathematically with the digital circuits, we need to use the Boolean Algebra. It can be defined with the help of the following:

1. A set of elements
2. A set of operators
3. A number of unproved postulates.

Boolean (Binary) Algebra:

It is used to analyze and simplify the digital (logic) circuits. Since it uses only the binary numbers i.e. 0 and 1 it is also called as "Binary Algebra", or "Logical Algebra".

The rules of Boolean Algebra are different from those of the conventional Algebra in the following manner;

1. Symbols used in Boolean Algebra (usually letters) do not represent numerical values.
2. Arithmetic operations (addition, subtraction, division etc.) are not performed in boolean algebra. Also there are no fractions, negative numbers, square, square root, logarithms, imaginary numbers etc.
3. Third and most important point is Boolean Algebra allows only two possible values ("0" to "1") for any variable.

Rules in Boolean algebra: There are some rules to be followed while while using a Boolean Algebra, these are:

1. Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
2. Complement of a variable is represented by a overbar (-). Thus complement of variable B is represented as B bar. Thus if A = 0 then Ā = 1 and if A = 1 then Ā = 0.
3. ORing of the variables is represented by a plus (+) sign between them. For example ORing of A, B, C is represented as A+B+C.
4. Logical ANDing of the two or more variables is represented by writing a dot between them such as A.B.C.D.E. Sometimes the dot may be omitted like ABCDE.

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BOOLEAN LAWS

Some basic Logic Gate laws;

OR operation

A + 0 = A

A + 1 = 1

A + A = A

A + Ā = 1

AND operation

A . 0 = 0

A . 1 = A

A . A = A

A + Ā = 1

NOT operation

A = Ā

Commutative Law

A + B = B + A

Associative Law

A + ( B + C ) = ( A + B ) + C

Distributive Law

A . ( B + C ) = A.B + A.C

De Moragan's Theorem

To obtain the inverse of any Boolean function, invert all variable and replace all OR's by AND's and vice versa.

Truth Table to Prove De Morgan's Theorem

Duality Theorem

Starting with a Boolean relation, we can derive another Boolean relation, called its dual by the following steps;

1. Changing each OR sign to an AND sign
2. Changing each AND sign to an OR sign
3. Complementing all 0's and 1's. Example; dual of A.0 = 0 is written as A+1=1.

Applying duality theorem to A.(B + C) = A.B + A.C by relationship we get;

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

Some other Boolean Expressions:

• A . ( B + C ) = A . B + A . C
• A + ( B . C ) = ( A + B )( A + C )
• A + ( A . B ) = A
• A . ( A + B ) = A
• A + Ā B = A + B
• A . ( Ā + B ) = A . B
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• A . B + Ā . C = ( A + C ) . ( Ā + B )
• ( A + B ) . (Ā + C ) = ( A . C ) + ( Ā . C )
• ( A . B ) + ( Ā . C ) + ( B . C ) = ( A . B ) + ( Ā . C )
• ( A + B ) . ( Ā + C ) . ( B + C ) = ( A + B ) . ( Ā + C )