As well as the logic symbols “0” and “1” being used to represent a digital input or output, we can also use them as constants for a permanently “Open” or “Closed” circuit or contact respectively.
A set of rules or Laws of Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean Algebra.
Boolean Algebra is the mathematics we use to analyse digital gates and circuits. We can use these “Laws of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required. Boolean Algebra is therefore a system of mathematics based on logic that has its own set of rules or laws which are used to define and reduce Boolean expressions.
The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labelled individually to represent inputs to the expression, For example, variables A, B, C etc, giving us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.
Examples of these individual laws of Boolean, rules and theorems for Boolean Algebra are given in the following table.
Law or Rule
|A + 1 = 1||A in parallel with
closed = "CLOSED"
|A + 0 = A||A in parallel with
open = "A"
|A . 1 = A||A in series with
closed = "A"
|A . 0 = 0||A in series with
open = "OPEN"
|A + A = A||A in parallel with
A = "A"
|A . A = A||A in series with
A = "A"
|NOT A = A||NOT NOT A
(double negative) = "A"
|A + A = 1||A in parallel with
NOT A = "CLOSED"
|A . A = 0||A in series with
NOT A = "OPEN"
|A+B = B+A||A in parallel with B =
B in parallel with A
|A.B = B.A||A in series with B =
B in series with A
|A+B = A.B||invert and replace OR with AND||de Morgan’s Theorem|
|A.B = A+B||invert and replace AND with OR||de Morgan’s Theorem|
The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a change in position for addition and multiplication, the Associative Law allowing the removal of brackets for addition and multiplication, as well as the Distributive Law allowing the factoring of an expression, are the same as in ordinary algebra.
Each of the Boolean Laws above are given with just a single or two variables, but the number of variables defined by a single law is not limited to this as there can be an infinite number of variables as inputs too the expression. These Boolean laws detailed above can be used to prove any given Boolean expression as well as for simplifying complicated digital circuits.
A brief description of the various Laws of Boolean are given below with A representing a variable input.
Other algebraic Laws of Boolean not detailed above include:
Using the information above, simple 2-input AND, OR and NOT Gates can be represented by 16 possible functions as shown in the following table.
|7.||A AND B (AND)||A . B|
|8.||A AND NOT B||A . B|
|9.||NOT A AND B||A . B|
|10.||NOT AND (NAND)||A . B|
|11.||A OR B (OR)||A + B|
|12.||A OR NOT B||A + B|
|13.||NOT A OR B||A + B|
|14.||NOT OR (NOR)||A + B|
|15.||Exclusive-OR||A . B + A . B|
|16.||Exclusive-NOR||A . B + A . B|
Using the above laws, simplify the following expression: (A + B)(A + C)
|Q =||(A + B).(A + C)|
|A.A + A.C + A.B + B.C||– Distributive law|
|A + A.C + A.B + B.C||– Idempotent AND law (A.A = A)|
|A(1 + C) + A.B + B.C||– Distributive law|
|A.1 + A.B + B.C||– Identity OR law (1 + C = 1)|
|A(1 + B) + B.C||– Distributive law|
|A.1 + B.C||– Identity OR law (1 + B = 1)|
|Q =||A + (B.C)||– Identity AND law (A.1 = A)|
Then the expression: (A + B)(A + C) can be simplified to A + (B.C) as in the Distributive law.