Introduction to Boolean Algebra
By Stephen Bucaro
Boolean algebra is a system of mathematics in which the values of the variables can
take on only one of two values, either 0 or 1 (sometimes referred to as false and true).
The name "Boolean" comes from the name of its inventor, George Boole. Similar to regular
algebra, Boolean algebra can be used to simplify a mathematical expression. And since
computer logic is also a system in which the values of the inputs and outputs can take on
only one of two values, Boolean algebra can be used to simplify logic circuits.
Boolean algebra consists of a set of postulates and a set of laws. I will
list the postulates and laws in the tables below. In Boolean algebra the plus (+) symbol
represents the logical OR operation, the dot (.) represents the logical AND operation,
the prime (') represents the compliment operation (e.g. changing 0 to 1 or changing 1 to 0).
I know that to someone unfamiliar with Boolean algebra, the tables below look scary
and boring, but you don't need to pay much attention to them at this point, I'm simply
introducing them. You'll understand them more easily after I give examples of the use
of some of these postulates and laws.
Boolean Algebra Postulates (sometimes called identities)
|A + 0 = A||Identity law||ORing a variable with 0 is always equal to the varable|
|A . 1 = A||Identity law||ANDing a variable with 1 is always equal to the varable|
|A + A = A||Identity law||ORing a variable with itself is always equal to that varable|
|A . A = A||Identity law||ANDing a varable with itself is always equal to that varable|
|A + 1 = 1||Identity law||ORing a variable with 1 always results in 1|
|A . 0 = 0||Identity law||ANDing a varaible with 0 always results in 0|
|A + A' = 1||Complement law||ORing a variable with its complement always equals 1|
|A . A' = 0||Complement law||ANDing a variable with its complement always equals 0|
|A'' = X||Complement law||A double complement of a variable is that variable|
Boolean Algebra Laws
|X+Y+Z = Z+Y+X||Commutative law of ORing||Changing the order of ORing variables does not change the result|
|X.Y.Z = Y.Z.X||Commutative law of ANDing||Changing the order of ORing variables does not change the result|
|X+(Y+Z) = (X+Y)+Z||Associative law of ORing||When ORing grouped variables changing the grouping does not change the result|
|X.(Y.X) = (X.Y).Z||Associative law of ANDing||When Anding grouped variables changing the grouping does not change the result|
|A(B+C) = AB+AC||Distributive law of ANDing over ORing||When ANDing a variable with a bracketed expression of ORed variables, you can remove the brackets and AND the variable with each individual ORed variable|
|A+(B.C) = (A+B).(A+C)||Distributive law of Oring over ANDing||When ORing a varaible with a bracketed expression of ANDed variables, you can remove the brackets and OR the variable with each individual ANDed variable|
|(X+Y)' = X'.Y'||de Morgan's law||The complement of Oring two variables is equivalent to complementing each variable and ANDing those two variables|
|(X.Y)' = X'+Y'||de Morgan's law||The complement of ANDing two variables is equivalent to complementing each variable and Oring those two variables|
|A+(A.B) = A||Redundance law||A combination of the result of using the Identity law and Distributive law|
|A.(A+B) = A||Redundance law||A combination of the result of using the Identity law and Distributive law|
Redundance law proof
A can be written as A.1 which the Identity law says equals 1 so A + AB can be written as A . (1+B).
The Identity law says 1 + B = 1, so A . (1+B) can be written as A . 1 which the Identity law says equals A.
In A . (A + B), the Distributive law says can be written as A.A + A.B and the Identity law says A . A = A,
so we get A + AB, which as shown above equals A.