Canonical Forms:There are two types of canonical forms:
Disjunctive Normal Forms or Sum of Products or (SOP):A Boolean expression over ({0, 1}, ∨,∧,') is said to be in disjunctive normal form if it is a join of minterms Example: (x_{1}'∧x_{2}'∧x_{3}')∨( x_{1}'∧x_{2}∧x_{3}' )∨(x_{1}∧x_{2}∧x_{3}) is a Boolean expression in disjunctive normal form. Since there are three minterms x_{1}'∧x_{2}'∧x_{3}',x_{1}'∧x_{2}∧x_{3} and x_{1}∧x_{2}∧x_{3}. Maxterm: A Boolean Expression of n variables x_{1},x_{2},....x_{n}is said to be a maxterm if it is of the form x_{1}∨x_{2}∨..........∨x_{n} where x_{i} is used to denote x_{i} or x_{i}'. Conjunctive Normal Forms or Products of Sums or (POS):A Boolean expression over ({0, 1}, ∨,∧,') is said to be in a disjunctive normal form if it is a meet of maxterms Example: (x_{1}∨x_{2}∨x_{3})∧( x_{1}∨x_{2}∨x_{3} )∧(x_{1}∨x_{2}∨x_{3} )∧(x_{1}'∨x_{2}∨x_{3}' )∧(x_{1}'∧x_{2}'∧x_{3}) is a Boolean expression in conjunctive normal form consisting of five maxterms. Obtaining A Disjunctive Normal Form:Consider a function from {0, 1}^{n} to {0, 1}. A Boolean expression can be obtained in disjunctive normal forms corresponding to this function by having a minterm corresponding to each ordered ntuples of 0's and 1's for which the value of the function is 1. Obtaining A Conjunctive Normal Form:Consider a function from {0, 1}^{n} to {0, 1}. A Boolean expression can be obtained in conjunctive normal forms corresponding to this function by having a maxterm corresponding to each ordered ntuples of 0's and 1's for which the value of function is0. Example: Express the following function in
Solution:
Principle of Duality:The dual of any expression E is obtained by interchanging the operation + and * and also interchanging the corresponding identity elements 0 and 1, in original expression E. Example: Write the dual of following Boolean expressions: 1. (x_{1}*x_{2}) + (x_{1}*x_{3}') 2. (1+x_{2})*( x_{1}+1) Solution: 1. ( x_{1}+x_{2})*( x_{1}+x_{3}') 2. (0*x_{2})+( x_{1}*0) Note: The dual of any theorem in a Boolean algebra is also a theorem.
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