The logical symbol 0 and 1 are used for representing the digital input or output. The symbols "1" and "0" can also be used for a permanently open and closed digital circuit. The digital circuit can be made up of several logic gates. To perform the logical operation with minimum logic gates, a set of rules were invented, known as the Laws of Boolean Algebra. These rules are used to reduce the number of logic gates for performing logic operations.
The Boolean algebra is mainly used for simplifying and analyzing the complex Boolean expression. It is also known as Binary algebra because we only use binary numbers in this. George Boole developed the binary algebra in 1854.
Rules in Boolean algebra
Properties of Boolean algebra
These are the following properties of Boolean algebra:
When the variable is AND with 0, it will give the result 0, and when the variable is OR with 1, it will give the result 1, i.e.,
B.0 = 0
B+1 = 1
When the variable is AND with 1 and OR with 0, the variable remains the same, i.e.,
B.1 = B
B+0 = B
When the variable is AND and OR with itself, the variable remains same or unchanged, i.e.,
B.B = B
B+B = B
When the variable is AND and OR with its complement, it will give the result 0 and 1 respectively.
B.B' = 0
B+B' = 1
Double Negation Law
This law states that, when the variable comes with two negations, the symbol gets removed and the original variable is obtained.
((A)')' = A
This law states that no matter in which order we use the variables. It means that the order of variables doesn't matter in this law.
A.B = B.A
A+B = B+A
This law states that the operation can be performed in any order when the variables priority is of same as '*' and '/'.
(A.B).C = A.(B.C)
(A+B)+C = A+(B+C)
This law allows us to open up of brackets. Simply, we can open the brackets in the Boolean expressions.
A+(B.C) = (A+B).(A+C
A.(B+C) = (A.B)+(A.C)
This law allows us for absorbing the similar variables.
B+(B.A) = B
B.(B+A) = B
De Morgan Law
The operation of an OR and AND logic circuit will remain same if we invert all the inputs, change operators from AND to OR and OR to AND, and invert the output.
(A.B)' = A'+B'
(A+B)' = A'.B'