Ref. Page Slide /78 Learning Objectives In this chapter you will learn about: oolean algebra Fundamental concepts and basic laws of oolean algebra oolean function and minimization Logic gates Logic circuits and oolean expressions ombinational circuits and design Ref. Page 6 Slide 2/78 oolean lgebra n algebra that deals with binary number system George oole (85-864), an English mathematician, developed it for: Simplifying representation Manipulation of propositional logic In 938, laude E. Shannon proposed using oolean algebra in design of relay switching circuits Provides economical and straightforward approach Used extensively in designing electronic circuits used in computers Ref. Page 6 Slide 3/78
Fundamental oncepts of oolean lgebra Use of inary igit oolean equations can have either of two possible values, and Logical ddition Symbol +, also known as OR operator, used for logical addition. Follows law of binary addition Logical Multiplication Symbol., also known as N operator, used for logical multiplication. Follows law of binary multiplication omplementation Symbol -, also known as NOT operator, used for complementation. Follows law of binary compliment Ref. Page 6 Slide 4/78 Operator Precedence Each operator has a precedence level Higher the operator s precedence level, earlier it is evaluated Expression is scanned from left to right First, expressions enclosed within parentheses are evaluated Then, all complement (NOT) operations are performed Then, all (N) operations are performed Finally, all + (OR) operations are performed Ref. Page 62 Slide 5/78 Operator Precedence X + Y Z st 2 nd 3 rd Ref. Page 62 Slide 6/78 2
Postulates of oolean lgebra Postulate : (a) =, if and only if, is not equal to (b) =, if and only if, is not equal to Postulate 2: (a) x + = x (b) x = x Postulate 3: ommutative Law (a) x + y = y + x (b) x y = y x Ref. Page 62 Slide 7/78 Postulates of oolean lgebra Postulate 4: ssociative Law (a) x + (y + z) = (x + y) + z (b) x (y z) = (x y) z Postulate 5: istributive Law (a) x (y + z) = (x y) + (x z) (b) x + (y z) = (x + y) (x + z) Postulate 6: (a) x + x = (b) x x = Ref. Page 62 Slide 8/78 The Principle of uality There is a precise duality between the operators. (N) and + (OR), and the digits and. For example, in the table below, the second row is obtained from the first row and vice versa simply by interchanging + with. and with olumn olumn 2 olumn 3 Row + = + = + = + = Row 2 = = = = Therefore, if a particular theorem is proved, its dual theorem automatically holds and need not be proved separately Ref. Page 63 Slide 9/78 3
Some Important Theorems of oolean lgebra Sr. No. Theorems/ Identities ual Theorems/ Identities Name (if any) x + x = x x x = x Idempotent Law 2 x + = x = 3 x + x y = x x x + y = x bsorption Law 4 x = x Involution Law 5 x x + y = x y x + x y = x + y 6 x+ y = x y x y = x y+ e Morgan s Law Ref. Page 63 Slide /78 Methods of Proving Theorems The theorems of oolean algebra may be proved by using one of the following methods:. y using postulates to show that L.H.S. = R.H.S 2. y Perfect Induction or Exhaustive Enumeration method where all possible combinations of variables involved in L.H.S. and R.H.S. are checked to yield identical results 3. y the Principle of uality where the dual of an already proved theorem is derived from the proof of its corresponding pair Ref. Page 63 Slide /78 Proving a Theorem by Using Postulates (Example) Theorem: x + x y = x Proof: L.H.S. = x + x y = x + x y by postulate 2(b) = x ( + y) by postulate 5(a) = x (y + ) by postulate 3(a) = x by theorem 2(a) = x by postulate 2(b) = R.H.S. Ref. Page 64 Slide 2/78 4
Proving a Theorem by Perfect Induction (Example) Theorem: x + x y = x = x y x y x + x y Ref. Page 64 Slide 3/78 Proving a Theorem by the Principle of uality (Example) Theorem: Proof: x + x = x L.H.S. = x + x = (x + x) by postulate 2(b) = (x + x) (x + X) by postulate 6(a) = x + x X by postulate 5(b) = x + by postulate 6(b) = x by postulate 2(a) = R.H.S. Ref. Page 63 Slide 4/78 Proving a Theorem by the Principle of uality (Example) ual Theorem: Proof: x x = x L.H.S. = x x = x x + by postulate 2(a) = x x + x X by postulate 6(b) = x (x + X) by postulate 5(a) = x by postulate 6(a) = x by postulate 2(b) = R.H.S. Notice that each step of the proof of the dual theorem is derived from the proof of its corresponding pair in the original theorem Ref. Page 63 Slide 5/78 5
oolean Functions oolean function is an expression formed with: inary variables Operators (OR, N, and NOT) Parentheses, and equal sign The value of a oolean function can be either or oolean function may be represented as: n algebraic expression, or truth table Ref. Page 67 Slide 6/78 Representation as an lgebraic Expression W = X + Y Z Variable W is a function of X, Y, and Z, can also be written as W = f (X, Y, Z) The RHS of the equation is called an expression The symbols X, Y, Z are the literals of the function For a given oolean function, there may be more than one algebraic expressions Ref. Page 67 Slide 7/78 Representation as a Truth Table X Y Z W Ref. Page 67 Slide 8/78 6
Representation as a Truth Table The number of rows in the table is equal to 2 n, where n is the number of literals in the function The combinations of s and s for rows of this table are obtained from the binary numbers by counting from to 2 n - Ref. Page 67 Slide 9/78 Minimization of oolean Functions Minimization of oolean functions deals with Reduction in number of literals Reduction in number of terms Minimization is achieved through manipulating expression to obtain equal and simpler expression(s) (having fewer literals and/or terms) Ref. Page 68 Slide 2/78 Minimization of oolean Functions F = x y z + x y z + x y F has 3 literals (x, y, z) and 3 terms F = x y + x z 2 F 2 has 3 literals (x, y, z) and 2 terms F 2 can be realized with fewer electronic components, resulting in a cheaper circuit Ref. Page 68 Slide 2/78 7
Minimization of oolean Functions x y z F F 2 oth F and F 2 produce the same result Ref. Page 68 Slide 22/78 Try out some oolean Function Minimization (a) x + x y ( ) (b) x x + y (c) x y z + x y z + x y (d) x y + x z + y z ( + ) ( + ) ( + ) (e) x y x z y z Ref. Page 69 Slide 23/78 omplement of a oolean Function The complement of a oolean function is obtained by interchanging: Operators OR and N omplementing each literal This is based on e Morgan s theorems, whose general form is: +++...+= 2 3 n 2 3... n 2 3...=+ n 2++...+ 3 n Ref. Page 7 Slide 24/78 8
omplementing a oolean Function (Example) F=xyz+xyz To obtain, we first interchange the OR and the N F operators giving ( x+y+z ) ( x+y+z) Now we complement each literal giving ( ) ( ) F=x+y+z x+y+z Ref. Page 7 Slide 25/78 anonical Forms of oolean Functions Minterms Maxterms : n variables forming an N term, with each variable being primed or unprimed, provide 2 n possible combinations called minterms or standard products : n variables forming an OR term, with each variable being primed or unprimed, provide 2 n possible combinations called maxterms or standard sums Ref. Page 7 Slide 26/78 x Variables y z Term Minterms and Maxterms for three Variables Minterms x y z x y z x y z x y z x y z x y z x y z x y z esignation m m m m m m m m Term Maxterms x + y + z x + y + z x + y + z x + y + z x + y + z x + y + z x + y + z x + y + z esignation Note that each minterm is the complement of its corresponding maxterm and vice-versa 2 3 4 5 6 7 M M M M M M M M 2 3 4 5 6 7 Ref. Page 7 Slide 27/78 9
Sum-of-Products (SOP) Expression sum-of-products (SOP) expression is a product term (minterm) or several product terms (minterms) logically added (ORed) together. Examples are: x x+yz xy+xy x+y xy+z xy+xyz Ref. Page 72 Slide 28/78 Steps to Express a oolean Function in its Sum-of-Products Form. onstruct a truth table for the given oolean function 2. Form a minterm for each combination of the variables, which produces a in the function 3. The desired expression is the sum (OR) of all the minterms obtained in Step 2 Ref. Page 72 Slide 29/78 Expressing a Function in its Sum-of-Products Form (Example) x y z F The following 3 combinations of the variables produce a :,, and Ref. Page 73 Slide 3/78
Expressing a Function in its Sum-of-Products Form (Example) Their corresponding minterms are: x y z, x y z, and x y z Taking the OR of these minterms, we get F=xy z+x y z+x y z=m +m + m F xy z =,4,7 4 7 ( ) ( ) Ref. Page 72 Slide 3/78 Product-of Sums (POS) Expression product-of-sums (POS) expression is a sum term (maxterm) or several sum terms (maxterms) logically multiplied (Ned) together. Examples are: x x+y ( x+y) z ( x+y) ( x+y) ( x+y) ( x+y)( x+y+z) ( x+y)( x+y) Ref. Page 74 Slide 32/78 Steps to Express a oolean Function in its Product-of-Sums Form. onstruct a truth table for the given oolean function 2. Form a maxterm for each combination of the variables, which produces a in the function 3. The desired expression is the product (N) of all the maxterms obtained in Step 2 Ref. Page 74 Slide 33/78
Expressing a Function in its Product-of-Sums Form x y z F The following 5 combinations of variables produce a :,,,, and Ref. Page 73 Slide 34/78 Expressing a Function in its Product-of-Sums Form Their corresponding maxterms are: ( ) ( ) ( ) ( x+y+z) and( x+y+z) x+y+z, x+y+z, x+y+z, Taking the N of these maxterms, we get: ( ) ( ) ( ) ( ) ( x+y+z) =M M2 M3 M5 M6 ( x,y,z) =Π(,2,3,5,6) F =x+y+z x+y+z x+y+z x+y+z F Ref. Page 74 Slide 35/78 onversion etween anonical Forms (Sum-of- Products and Product-of-Sums) To convert from one canonical form to another, interchange the symbol and list those numbers missing from the original form. Example: ( ) ( ) ( ) ( ) ( ) ( ) F x,y,z =Π,2,4,5 =Σ,3,6,7 F x,y,z =Σ,4,7 =Σ,2,3,5,6 Ref. Page 76 Slide 36/78 2
Logic Gates Logic gates are electronic circuits that operate on one or more input signals to produce standard output signal re the building blocks of all the circuits in a computer Some of the most basic and useful logic gates are N, OR, NOT, NN and NOR gates Ref. Page 77 Slide 37/78 N Gate Physical realization of logical multiplication (N) operation Generates an output signal of only if all input signals are also Ref. Page 77 Slide 38/78 N Gate (lock iagram Symbol and Truth Table) = Inputs Output = Ref. Page 77 Slide 39/78 3
OR Gate Physical realization of logical addition (OR) operation Generates an output signal of if at least one of the input signals is also Ref. Page 77 Slide 4/78 OR Gate (lock iagram Symbol and Truth Table) = + Inputs Output = + Ref. Page 78 Slide 4/78 NOT Gate Physical realization of complementation operation Generates an output signal, which is the reverse of the input signal Ref. Page 78 Slide 42/78 4
NOT Gate (lock iagram Symbol and Truth Table) Input Output Ref. Page 79 Slide 43/78 NN Gate omplemented N gate Generates an output signal of: if any one of the inputs is a when all the inputs are Ref. Page 79 Slide 44/78 NN Gate (lock iagram Symbol and Truth Table) = ==+ Inputs Output =+ Ref. Page 79 Slide 45/78 5
NOR Gate omplemented OR gate Generates an output signal of: only when all inputs are if any one of inputs is a Ref. Page 79 Slide 46/78 NOR Gate (lock iagram Symbol and Truth Table) = =+ = Inputs Output = Ref. Page 8 Slide 47/78 Logic ircuits When logic gates are interconnected to form a gating / logic network, it is known as a combinational logic circuit The oolean algebra expression for a given logic circuit can be derived by systematically progressing from input to output on the gates The three logic gates (N, OR, and NOT) are logically complete because any oolean expression can be realized as a logic circuit using only these three gates Ref. Page 8 Slide 48/78 6
Finding oolean Expression of a Logic ircuit (Example ) NOT = ( + ) OR + N Ref. Page 8 Slide 49/78 Finding oolean Expression of a Logic ircuit (Example 2) OR + ( ) ( ) = + N N NOT Ref. Page 8 Slide 5/78 onstructing a Logic ircuit from a oolean Expression (Example ) oolean Expression = + N OR + Ref. Page 83 Slide 5/78 7
onstructing a Logic ircuit from a oolean Expression (Example 2) oolean Expression = ++EF N NOT N N ++EF E F N EF NOT EF Ref. Page 83 Slide 52/78 Universal NN Gate NN gate is an universal gate, it is alone sufficient to implement any oolean expression To understand this, consider: asic logic gates (N, OR, and NOT) are logically complete Sufficient to show that N, OR, and NOT gates can be implemented with NN gates Ref. Page 84 Slide 53/78 Implementation of NOT, N and OR Gates by NN Gates =+= (a) NOT gate implementation. = (b) N gate implementation. Ref. Page 85 Slide 54/78 8
Implementation of NOT, N and OR Gates by NN Gates = = (c) OR gate implementation. = += + Ref. Page 85 Slide 55/78 Method of Implementing a oolean Expression with Only NN Gates Step : From the given algebraic expression, draw the logic diagram with N, OR, and NOT gates. ssume that both the normal () and complement ( ) inputs are available Step 2: raw a second logic diagram with the equivalent NN logic substituted for each N, OR, and NOT gate Step 3: Remove all pairs of cascaded inverters from the diagram as double inversion does not perform any logical function. lso remove inverters connected to single external inputs and complement the corresponding input variable Ref. Page 85 Slide 56/78 oolean Expression = Implementing a oolean Expression with Only NN Gates (Example) + ( + ) + ( + ) + ( + ) (a) Step : N/OR implementation Ref. Page 87 Slide 57/78 9
Implementing a oolean Expression with Only NN Gates (Example) N 5 OR 2 N 3 OR + ( + ) + N 4 ( + ) (b) Step 2: Substituting equivalent NN functions Ref. Page 87 Slide 58/78 Implementing a oolean Expression with Only NN Gates (Example) 2 5 ( +) + 3 4 (c) Step 3: NN implementation. Ref. Page 87 Slide 59/78 Universal NOR Gate NOR gate is an universal gate, it is alone sufficient to implement any oolean expression To understand this, consider: asic logic gates (N, OR, and NOT) are logically complete Sufficient to show that N, OR, and NOT gates can be implemented with NOR gates Ref. Page 89 Slide 6/78 2
Implementation of NOT, OR and N Gates by NOR Gates + = = (a) NOT gate implementation. + + = + (b) OR gate implementation. Ref. Page 89 Slide 6/78 Implementation of NOT, OR and N Gates by NOR Gates + = += = + = (c) N gate implementation. Ref. Page 89 Slide 62/78 Method of Implementing a oolean Expression with Only NOR Gates Step : For the given algebraic expression, draw the logic diagram with N, OR, and NOT gates. ssume that both the normal ( ) and complement ( ) inputs are available Step 2: raw a second logic diagram with equivalent NOR logic substituted for each N, OR, and NOT gate Step 3: Remove all parts of cascaded inverters from the diagram as double inversion does not perform any logical function. lso remove inverters connected to single external inputs and complement the corresponding input variable Ref. Page 89 Slide 63/78 2
Implementing a oolean Expression with Only NOR Gates (Examples) oolean Expression + + = + ( ) ( + ) (a) Step : N/OR implementation. ( + ) + Ref. Page 9 Slide 64/78 Implementing a oolean Expression with Only NOR Gates (Examples) 2 N N 3 OR + + 5 6 OR + ( ) (b) Step 2: Substituting equivalent NOR functions. 4 N ( + ) Ref. Page 9 Slide 65/78 Implementing a oolean Expression with Only NOR Gates (Examples) 2 5 6 3 4 (c) Step 3: NOR implementation. ( +) + Ref. Page 9 Slide 66/78 22
Exclusive-OR Function = + = =+ = =+ lso, ( ) = ( ) = Ref. Page 9 Slide 67/78 Exclusive-OR Function (Truth Table) Inputs Output = Ref. Page 92 Slide 68/78 Equivalence Function with lock iagram Symbol = + = = + lso, ( ) = ( ) = Ref. Page 9 Slide 69/78 23
Equivalence Function (Truth Table) Inputs Output = Ref. Page 92 Slide 7/78 Steps in esigning ombinational ircuits. State the given problem completely and exactly 2. Interpret the problem and determine the available input variables and required output variables 3. ssign a letter symbol to each input and output variables 4. esign the truth table that defines the required relations between inputs and outputs 5. Obtain the simplified oolean function for each output 6. raw the logic circuit diagram to implement the oolean function Ref. Page 93 Slide 7/78 esigning a ombinational ircuit Example Half-dder esign Inputs Outputs S S=+ = oolean functions for the two outputs. Ref. Page 93 Slide 72/78 24
esigning a ombinational ircuit Example Half-dder esign S=+ = Logic circuit diagram to implement the oolean functions Ref. Page 94 Slide 73/78 esigning a ombinational ircuit Example 2 Full-dder esign Inputs Truth table for a full adder Outputs S Ref. Page 94 Slide 74/78 esigning a ombinational ircuit Example 2 Full-dder esign oolean functions for the two outputs: S=+++ =+++ =++ (when simplified) Ref. Page 95 Slide 75/78 25
esigning a ombinational ircuit Example 2 Full-dder esign S (a) Logic circuit diagram for sums Ref. Page 95 Slide 76/78 esigning a ombinational ircuit Example 2 Full-dder esign (b) Logic circuit diagram for carry Ref. Page 95 Slide 77/78 Key Words/Phrases bsorption law N gate ssociative law oolean algebra oolean expression oolean functions oolean identities anonical forms for oolean functions ombination logic circuits umulative law omplement of a function omplementation e Morgan s law istributive law ual identities Equivalence function Exclusive-OR function Exhaustive enumeration method Half-adder Idempotent law Involution law Literal Logic circuits Logic gates Logical addition Logical multiplication Maxterms Minimization of oolean functions Minterms NN gate NOT gate Operator precedence OR gate Parallel inary dder Perfect induction method Postulates of oolean algebra Principle of duality Product-of-Sums expression Standard forms Sum-of Products expression Truth table Universal NN gate Universal NOR gate Ref. Page 97 Slide 78/78 26