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Boolean algebra is a system of logic dealing with variables that can have only two values:
True (represented by 1) and False (represented by 0). It provides a framework for reasoning and simplifying logical expressions, similar to how regular algebra works with numbers. Analysing digital gates and circuits is done with it. Performing a mathematical operation on binary numbers, or "0" and "1," makes sense. Basic operators in Boolean algebra include AND, OR, NOT, and so forth. The symbols "." for AND and "+" for OR denote operations. Variables that are represented by capital letters, such as "A," "B," and so forth, can be subjected to operations.
Properties of Boolean algebra –
Annulment law – a variable ANDed with 0 gives 0, while a variable ORed with 1 gives 1,
e.g.
A.0 = 0
A + 1 = 1
Identity law – in this law variable remains unchanged it is ORed with ‘0’ or ANDed with ‘1’,
e.g.
A.1 = A
A + 0 = A
Idempotent law – a variable remains unchanged when it is ORed or ANDed with itself,
e.g.
A + A = A
A.A = A
Complement law – in this Law, if a complement is added to a variable it gives one, if a variable is multiplied by its complement it results in ‘0
e.g.
A + A' = 1
A.A' = 0
Double negation law – a variable with two negations, its symbol gets cancelled out and the original variable is obtained,
e.g.
((A)')' = A
Commutative law – a variable order does not matter in this law
e.g.
A + B = B + A
A.B = B.A
Associative law – the order of operation does not matter if the priority of variables are the same like ‘*’ and ‘/’
e.g.
A+(B+C) = (A+B)+C
A.(B.C) = (A.B).C
Distributive law – this law governs the opening up of brackets
e.g.
A.(B+C) = (A.B)+(A.C)
(A+B)(A+C) = A + BC
Absorption law –This law involves absorbing similar variables,
e.g.
A.(A+B) = A
A + AB = A
A+ A'B = A+B
A(A' + B) = AB
De Morgan law – the operation of an AND or OR logic circuit is unchanged if all inputs are inverted, the operator is changed from AND to OR, and the output is inverted,
e.g.
(A.B)' = A' + B'
(A+B)' = A'.B
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