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The Fundamentals of Boolean Algebra: How to Simplify and Solve Logical Expressions

kartik yadav


Logic gate expression implementation
Logic gate expression implementation

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,