Question

Prove the following identities with the help of truth table.(i)(A+B+C)' = A'.B'.C' (ii)A.(A'+B) = A.B (iii)A.B+A'.C + B.C = A.B+A'.C

Answer 1

Answer:

Explanation:

Laws of Boolean Algebra

Boolean Algebra uses a set of Laws and Rules to define the operation of a digital logic circuit

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 A . 0 = 0    A variable AND’ed with 0 is always equal to 0

A + 1 = 1    A variable OR’ed with 1 is always equal to 1

 

Identity Law – A term OR‘ed with a “0” or AND‘ed with a “1” will always equal that term

 

A + 0 = A   A variable OR’ed with 0 is always equal to the variable

A . 1 = A    A variable AND’ed with 1 is always equal to the variable

 

Idempotent Law – An input that is AND‘ed or OR´ed with itself is equal to that input

 

A + A = A    A variable OR’ed with itself is always equal to the variable

A . A = A    A variable AND’ed with itself is always equal to the variableare given in the following table.