12/31/2010. Overview. 05-Boolean Algebra Part 3 Text: Unit 3, 7. DeMorgan s Law. Example. Example. DeMorgan s Law

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1 Overview 05-oolean lgebra Part 3 Text: Unit 3, 7 EEGR/ISS 201 Digital Operations and omputations Winter 2011 DeMorgan s Laws lgebraic Simplifications Exclusive-OR and Equivalence Functionally omplete NND-NOR gates Dr. Louie 2 DeMorgan s Law DeMorgan s Laws are helpful when complements appear in expressions: (+) = () = + Use a truth table to show (+) = + ( +) Dr. Louie 3 Dr. Louie 4 Use a truth table to show (+) = Is = ()? DeMorgan s Law + ( +) Dr. Louie 5 Dr. Louie 6 1

2 Is = ()? NO! DeMorgan s Law () DeMorgan s Laws an be generalized to n variables: ( 1 + n ) = 1 n ( 1 n ) = 1 + n The complement of the sum is the product of the complements The complement of the product is the sum of the complements Dr. Louie 7 Dr. Louie 8 DeMorgan s Laws Find the complement of ( +) DeMorgan s Laws Find the complement of ( +) (+) = () = + (( +) ) = ( +) + = + First set ( +) = ; = Then set = ; = Dr. Louie 9 Dr. Louie 10 omplete the truth table for: + Z +Z Z Z Z + Z + Z +Z onsider: + Z +Z Z Z Z + Z + Z +Z Identical columns Dr. Louie 11 Dr. Louie 12 2

3 In + Z +Z, the Z is redundant Given two pairs (in this case and Z) for which one variable appears in one term and it s complement in the other ( in this case), then the consensus term is found by multiplying the two original terms together See proof (page 63) Formally + Z + Z = + Z : identify, and Z in D + + D Dr. Louie 13 Dr. Louie 14 Formally + Z + Z = + Z : identify, and Z in D + + D = = D Z = D + + D = D + Note: usually you will not see D (instead: D), but you can still use the theorem lgebraic Simplifications Options are: 1. ombine Terms + = 2. Eliminate Terms + = 3. Eliminate Literals + = + 4. dd Redundant Terms dd Multiply by ( + ) dd Z to + Z (consensus) Dr. Louie 15 Dr. Louie 16 D + D + D + D + D+D + D D + D + D + D + D+D + D combining terms: D + D = D D + D + D + D+D + D eliminating terms: D + D = D Dr. Louie 17 Dr. Louie 18 3

4 D + D + D+D + D D + D( +) + D + D eliminating literals: + = + D + D( + ) + D + D D + D + D + D + D adding redundant terms: D + D = D + D + D + D + D + D + D + D + D + D + D + D + consensus: D + + D = D + D + D + D + D + consensus: D + + D D + D + D + = D + Dr. Louie 19 Dr. Louie 20 There are a number of ways to prove that D + D + D + and D + D + D + D + D+D + D are equivalent expressions In ordinary algebra we can show equivalence x + y = x + z is the same as y = z by subtracting x from both sides: x + y x = x + z x => y = z cancellation does not work in oolean algebra + = + Z does imply that = Z Let = 1, = 0, Z = = (but does not equal Z) Dr. Louie 21 Dr. Louie 22 Subtraction is not defined in oolean algebra Division is not defined either: xy = xz implies y = z (in ordinary algebra, so long as x does not equal zero) = Z does not imply = Z in oolean algebra will be equal to 0 about half the time 1. onstruct a truth table and check for identical results to all input combinations 2. Manipulate one side using various theorems until it is identical to the other side 3. Reduce both sides independently to the same expression 4. Perform reversible operations to both sides until equivalent Dr. Louie 23 Dr. Louie 24 4

5 Exclusive-OR Exclusive-OR Exclusive OR ( ), OR, 0 OR 0 = 0 0 OR 1 = 1 1 OR 0 = 1 1 OR 1 = 0 How can we construct the OR with NDs, ORs and NOTs? Exclusive OR ( ), OR, 0 OR 0 = 0 0 OR 1 = 1 1 OR 0 = 1 1 OR 1 = 0 How can we construct the OR with NDs and ORs? = + Dr. Louie 25 Dr. Louie 26 Exclusive OR Theorems 0 = 1 = = 0 = 1 = ( ) = ( ) ( Z) = Z ( ) = = = + Equivalence Operation Equivalence operation:, ompares values, if equal, then 1, else 0 (0 0) = 1 (0 1) = 0 (1 0) = 0 (1 1) = 1 Equivalence is the opposite of OR Equivalence is commutative and associative Dr. Louie 27 Dr. Louie 28 NND gate is really an ND-NOT gate Truth table is the complement of an ND gate NOR gate is really an OR-NOT gate Truth table is the complement of an OR gate Dr. Louie 29 Dr. Louie 30 5

6 Functionally omplete set of logic operations (e.g. ND, OR, NOT) are functionally complete if and oolean function can be expressed in terms of this set of operations The set ND, OR and NOT is functionally complete The set of ND and NOT is functionally complete OR gate can be constructed from them Derive NOT using NND gate(s) + Dr. Louie 31 Dr. Louie 32 Derive NOT using NND gate(s) = () = + ; = = = + = Derive ND using NND gate(s) Dr. Louie 33 Dr. Louie 34 Derive ND using NND gate(s) Z = (W) ; W = () Z = (() ) = ( + ) = Derive OR using NND gate(s) W Z W Z Dr. Louie 35 Dr. Louie 36 6

7 Derive OR using NND gate(s) Z = (UW) = (U W ) = U + W U W Z We can show that NOR is also functionally complete U W Z Dr. Louie 37 Dr. Louie 38 7

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