oolean lgebra
Introduction 1854: Logical algebra was published by George oole known today as oolean lgebra It s a convenient way and systematic way of expressing and analyzing the operation of logic circuits. 1938: Claude Shannon was the first to apply oole s work to the analysis and design of logic circuits.
oolean Operations & Expressions Variable a symbol used to represent a logical quantity. Complement the inverse of a variable and is indicated by a bar over the variable. Literal a variable or the complement of a variable.
oolean ddition oolean addition is equivalent to the OR operation 00 0 01 1 10 1 11 1 sum term is produced by an OR operation with no ND ops involved. i.e.,, C, C D sum term is equal to 1 when one or more of the literals in the term are 1. sum term is equal to 0 only if each of the literals is 0.
oolean Multiplication oolean multiplication is equivalent to the ND operation 0 0 0 0 1 0 1 0 0 1 1 1 product term is produced by an ND operation with no OR ops involved. i.e.,, C, CD product term is equal to 1 only if each of the literals in the term is 1. product term is equal to 0 when one or more of the literals are 0.
Laws & Rules of oolean lgebra The basic laws of oolean algebra: The commutative laws (กฏการสล บท )บท ) กฏการสล บท )บท )) The associative laws (กฏการจ ดกล มดกล บท )ม กฏการจ ดกล มดกล บท )ม) The distributive laws (กฏการกระจ ดกล มายกล มาย)
Commutative Laws The commutative law of addition for two variables is written as: The commutative law of multiplication for two variables is written as:
ssociative Laws The associative law of addition for 3 variables is written as: (C) ()C C C (C) The associative law of multiplication for 3 variables is written as: (C) ()C C ()C C C (C) C ()C
Distributive Laws The distributive law is written for 3 variables as follows: (C) C C C X C C X X(C) XC
Rules of oolean lgebra 1. 0 7. 2. 1 1 8. 0 3. 0 4. 1 0 9. 10. 5. 11. 6. 1 12.( )( C) C,, and C can represent a single variable or a combination of variables.
DeMorgan s Theorems DeMorgan s theorems provide mathematical verification of: the equivalency of the NND and negative-or gates the equivalency of the NOR and negative-nd gates.
DeMorgan s Theorems The complement of two or more NDed variables is equivalent to the OR of the complements of the individual variables. NND Negative-OR X Y X Y The complement of two or more ORed variables is equivalent to the ND of the complements of the individual variables. NOR X Y Negative-ND X Y
DeMorgan s Theorems (Exercises) pply DeMorgan s theorems to the expressions: X Y Z X Y Z X Y Z W X Y Z
DeMorgan s Theorems (Exercises) pply DeMorgan s theorems to the expressions: ( C) D C DEF CD EF C D( E F )
oolean nalysis of Logic Circuits oolean algebra provides a concise way to express the operation of a logic circuit formed by a combination of logic gates so that the output can be determined for various combinations of input values.
oolean Expression for a Logic Circuit To derive the oolean expression for a given logic circuit, begin at the left-most inputs and work toward the final output, writing the expression for each gate. C D CD CD (CD)
Constructing a Truth Table for a Logic Circuit Once the oolean expression for a given logic circuit has been determined, a truth table that shows the output for all possible values of the input variables can be developed. Let s take the previous circuit as the example: (CD) There are four variables, hence 16 (2 4 ) combinations of values are possible.
Constructing a Truth Table for a Logic Circuit Evaluating the expression To evaluate the expression (CD),, first find the values of the variables that make the expression equal to 1 (using the rules for oolean add & mult). In this case, the expression equals 1 only if 1 and CD1 because (CD) 1 11 1 1
Constructing a Truth Table for a Logic Circuit Evaluating the expression (cont ) Now, determine when CD term equals 1. The term CD1 if either 1 or CD1 or if both and CD equal 1 because CD 10 1 CD 01 1 CD 11 1 The term CD1 only if C1 and D1
Constructing a Truth Table for a Logic Circuit Evaluating the expression (cont ) Summary: (CD)1 When 1 and 1 regardless of the values of C and D When 1 and C1 and D1 regardless of the value of The expression (CD)0 for all other value combinations of the variables.
Constructing a Truth Table for a Logic Circuit Putting the results in truth table format (CD)1 When 1 and 1 regardless of the values of C and D When 1 and C1 and D1 regardless of the value of INPUTS OUTPUT C D (CD) 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1