Introduction to Boolean Algebra
Boolean Algebra is a kind of algebra that is generated by operating the binary system. It was introduced by George Boole in 1854. He was an English mathematician, who primarily worked on differential equations and the calculus of variations. But the work for which he is best known is – application of algebraic systems to non-mathematical reasoning. He proposed that logical propositions should be expressed as algebraic equations, so now logic was reduced to algebra. In his logical deductions he replaced the operation of multiplication by AND, and addition by OR.
Boolean Algebra is made up of:
• Elements – which are variables or constants with value 1 or 0.
• Operators – which are And, Or and Not.
• Axioms and Theorems.
A boolean variable is a symbol used to represent a logical quantity. It will take value from the domain {0, 1}, and boolean constant is single digit binary value (bit) viz. 0 or 1.
There are three fundamental operators: AND, OR and NOT.
• AND is a binary operator, to perform logical multiplication, it is represented by ‘.’
• OR is also a binary operator, to perform logical addition. It is represented by ‘+’.
• NOT is a unary operator, to complement the operand. Not is represented as ‘ or ¯.
Complement is the inverse of a variable/ constant. In case of boolean algebra, since the variable/constant can have value 0 or 1 so complement will be 1 or 0.
Just like algebra, Boolean algebra also have axioms and theorems, which describe how logical quantities behave. We know that axiom is a statement which is considered to be true, and theorems are to be proved. First axiom is Closure Property, it states that Boolean operations (+ or .) on Boolean variables or constants will always result into a Boolean value.
Other axioms and theorems of Boolean Algebra:
| Complement: |
| • Complement states that sum of a Boolean quantity with its complement or product of a Boolean quantity with its complement results into identity.
A + A’ = 1 and A . A’ = 0 |
| Identity: |
| • Identity states that, sum of anything and zero, or product of anything and one, is same as the original anything. So identity with respect to + is 0, and with respect to . is 1.
A + 0 = A and A.1 = A |
| Distributive: |
| • Distributive law states that, ORing variables and then ANDing the result with single variable is equivalent to ANDing the result with a single variable with each of the several variables and then ORing the products. Vice versa with respect to operators and terms is also true.
A . ( B + C ) = A . B + A . C and A + ( B . C ) = ( A + B ) . ( A + C ) |
| Commutative: |
| • Commulative property tells us, we can reverse the order of variables, that are either ORed together or ANDed together without changing the truth of the expression.
A + B = B + A and A . B = B . A |
| Indempotency: |
| • Indempotency states that when we sum or product a boolaen quantity to itself, the resultant is original quantity.
A + A = A and A . A = A |
| Null Element: |
| • No matter what the value of Boolean variable, the sum of variable and 1 will always be 1. Also the product of Boolean variable and 0 will always be 0.
A + 1 = 1 and A . 0 = 0 |
| Absorption: |
| • Absorption is also known as covering, have three forms:
1. A + AB = A and A . (A + B) = A |
| Associative: |
| • Associative property tells us that we can associate, group of sum or product variables together with parenthesis without altering the truth of the expression.
A + ( B + C ) = ( A + B ) + C and A . ( B . C ) = ( A . B) . C |
| Involution: |
| • Involution states that complementing a variable twice, or any even number of times, results in the original Boolean value.
(A’)’ = A |
| De Morgan’s Theorem: |
| • De Morgan’s Theorem states that the complement of a sum / product, equals the product / sum of the complements.
( A + B ) ‘ = A’ . B’ and ( A . B )’ = A’ + B’ |
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