Page 33 - Handout Digital Electronics

4.2 Laws of Boolean algebra

The basic laws/properties of Boolean algebra are:

• Commutative
• Distributive
• Identity
• Complement
• Associative

The commutative property states that: The order that two Boolean variables appear in an AND or function
is not significant, for example (AB = BA), (A+B) = (B+A).

The distributive property shows how a Boolean variable is distributed over an expression with which it is
ANDed, for example A (B+C) = AB+AC or A+BC= (A+B) (A+C).

The identity property states that a variable that is Ande or is ORed with itself produces the original
variable, for example A.A = A or A + A = A = A.

The complement is derived from the involution theorem which states that the complement of a
complement leaves the original variable unchanged.

A A
For example:
A A

The Associative law/property states that the order of ORing or ANDing Boolean variables is logically of
no consequence, for example (A+B) +C =A+ (B+C) or (A.B) C = A (BC).

4.3 The De Morgan’s Theorem

This theorem has the most significance in that it is a technique for substituting AND operators for OR
operators and vice-versa OR NOR for NAND function using group complementation. For example, the
logic function A+B when subjected to De Morgan theorem forms the equality:

A B A.B or AB A B

Group complementation means a long bar extending over two or more variables, for example:

AB The bar (-) above AB is a group complementation because it extends over two Boolean variables.

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