DLD: Laws of Boolean Algebra

Prepared by:Saad Aslam

Boolean Operations and Expressions

Boolean Algebra is the mathematics of digital system .A basic knowledge of Boolean Algebra is the study and analysis of logic circuits .Variables and complements terms are used in this chapter.

A variable can have  a 1 or 0 value.

A complement is the inverse of the variable indicated by a bar over the variable .For example:

Complement of variable A is A.If A is 0 the A is 1 and if A is 1 then A  is 0.The complement of a variable is read as “not A” or “A bar”.

Laws and Rules of Boolean Algebra

Laws of Boolean Algebra:

The basics Laws of Boolean Algebra are as follows:

·         Commutative Laws

·         Associative Laws

·         Distributive Laws

Commutative Laws

For addition

A+B=B+A

The figure below illustrates the Commutative law as applied to OR gates.

For Multiplication

AB=BA

The figure below illustrates the Commutative law as applied to AND gates.

Associative Laws

For Addition

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

The figure below illustrates the Associative law as applied to OR gates.

For Multiplication

A(BC)=(AB)C

The figure below illustrates the Associative law as applied to AND gates.

Distributive Laws

A(B+C)=AB + BC

The figure below illustrates the Distributive Law by using both AND ,OR gates.

Rules of Boolean Algebra:

Basic rules that are useful in manipulating and simplifying Boolean Expressions are as follows:

Rule 1: A+0=A

A variable ORed with 0 will always equal to the variable.

Rule 2 :A+1=1

A variable ORed with 1 will always equal to 1.

Rule 3 : A.0=0

A variable ANDed with 0 will always equal to 0.

Rule 4: A.1=A

A variable ANDed with 1 will always equal to variable.

Rule 5: A+A=A

A variable ORed with itself will always equal to the variable.

Rule 6: A+A=1

A variable ORed with the complement of itself will equal to 1.

Rule 7: A.A=A

A variable ANDed with itself will always equal to the variable.

Rule 8: A.A=0

A variable ANDed with the complement of itself will always equal to 0.

 

Rule 9 : A =A

A variable by complementing itself twice will equal to the variable.

Rule 10: A+ AB=A

Proof

A+AB=A(1+B)                                 Distributive Law

        =A.1                                         Rule 2 (1+B=1)

        =A                                            Rule 4 (A.1=A)

Rule 11 : A+AB=A+B

Proof

A + AB= (A+AB)+ AB                     Rule 10 A+ AB=A

        = (AA+AB)+ AB                      Rule 7 A.A=A

        = AA+AB+AA+ AB                 Rule 8  A.A=0

       =(A+A)(A+B)

       =1.(A+B)                                 Rule 6  A+A=1

       =A+B                                       Rule 4  A.1=A