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: (x1'∧x2'∧x3')∨( x1'∧x2∧x3' )∨(x1∧x2∧x3) is a Boolean expression in disjunctive normal form. Since there are three min-terms x1'∧x2'∧x3',x1'∧x2∧x3 and x1∧x2∧x3. Max-term: A Boolean Expression of n variables x1,x2,....xnis said to be a max-term if it is of the form x1∨x2∨..........∨xn where xi is used to denote xi or xi'. 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 max-terms Example: (x1∨x2∨x3)∧( x1∨x2∨x3 )∧(x1∨x2∨x3 )∧(x1'∨x2∨x3' )∧(x1'∧x2'∧x3) is a Boolean expression in conjunctive normal form consisting of five max-terms. 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 min-term corresponding to each ordered n-tuples 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 max-term corresponding to each ordered n-tuples 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. (x1*x2) + (x1*x3') 2. (1+x2)*( x1+1) Solution: 1. ( x1+x2)*( x1+x3') 2. (0*x2)+( x1*0) Note: The dual of any theorem in a Boolean algebra is also a theorem.Next TopicLogic Gates and Circuits |