Basic Boolean Laws

BASIC BOOLEAN LAWS

Logical operations can be expressed and minimized mathematically using the rules, laws and theorems of boolean algebra.

Boolean Addition:

Boolean addition involves variables having values of either a binary 1 or 0. The basic rule for addition are given below

0+0=0

0+1=1

1+0=1

1+1=1

Boolean addition is same as the logical OR operation Boolean multiplication. Boolean multiplication involves logical AND operation.

0.0 = 0

0.1 = 0

1.0 = 0

1.1 = 1

The most common postulates used to formulate various algebraic structures are

1. Closure:

A set S is closed with respect to a binary operates if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S.

For example, the set of natural numbers N = {1, 2, 3, 4 ...} is closed with respect to the binary operator plus (+) by the rules of arithmetic addition, since for any a, b, EN we obtain a unique C ∈ N by the operation a + b = c.

2. Associative low:

The associative property for addition is given by

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

The OR operation of several variables results in the same, regardless of the grouping of the variables.

The associative law of multiplication is given by

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

According to this law, it makes no difference in what order the variables are grouped during the AND operation of several variables.

Example

Law I→ A+ (B+ C) = (A+B) + C

From the above example grouping of the variable regardless of the grouped output is same Left hand side A+ (B+ C) equal to Right hand side (A + B) + C.

Law II → A.(B-C) = (A.B).C

From the above example A (B.C) = (A.B). C are same

Commutative Law

According to this law, the Boolean addition is commutative is given by

A+ B = B+ A

The OR operation conducted on the variables makes no difference.

A.B = B.A

This means AND operation conducted on the variables makes no difference.

Distributive Law

The boolean addition is distributive over boolean multiplications, given by

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

This law states that the AND operation of several variables and then the OR operation of the result with a single variable is equivalent to the OR operation of single variable with each of the several variables and then the AND operation of the sums.

Example

A(B + C) = AB+ AC