## Digital Electronics : Boolean Algebra And Logic Gates

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• ### Parag P

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This is the third tutorial in the digital electronics tutorial series and it introduces and discusses the type of algebra used to deal with digital logic circuits called as Boolean algebra. It emphasizes upon different boolean laws and boolean algebric manipulations in terms of Sum of Product (SoP) and Product of Sum (PoS) forms and other logic operations.

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Digital Electronics Presentation on Lecture 3 : Boolean Alzebra and Logic Gates Presented By : Parag Parandkar Assistant Professor, ECE 1
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Acknowledgement The presenter would like to thanks and acknowledge for the adoption of slides from the slides prepared by Dr. Rao and Mr. Jeevan Reddy. The copyrights belongs to the original author. The presentation is being used for educational and non commercial purpose. The presentation is being designed from the Digital Design by Morrois Mano
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Boolean Algebra
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Contents Boolean laws & Properties Boolean alzebric manipulations SOP and POS Forms Other logic operations
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Symbolic Logic Boolean algebra derives its name from the Mathematician George Boole. Symbolic Logic uses values, variables and operations : True is represented by the value 1. False is represented by the value 0. Variables are represented by letters and can have one of two values, either 0 or l. Operations are functions of one or more variables. AND is represented by X. Y OR is represented by X + Y NOT is represented by X' Throughout this tutorial the X' form will be used and sometimes !X will be used.
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These basic operations can be combined to give expressions. Examples : x relates to a statement (T/F) relates to two statements. True only when both the statements are true. W.X.Y + Z relates to four statements. The expression is true when W, X and Y are all true or when Z alone is true.
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Precedence As with any other branch of mathematics, these operators have an order of precedence. NOT operations have the highest precedence, followed by ND operations, followed by R operations. Brackets can be used as with other forms of algebra. e.g. XX + Z and are not the same function.
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Exercises in Precedence: If X, Y and Z are 0,1,1 respectively find the result of the following expressions : x'+ y',z + x,z' + z,y') + x,(y,(z+(x,y,(z+y') + z)))
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Function Definitions The logic operations given previously are defined as follows . Define f(X,Y) to be some function of the variables X and Y. = X.Y 1 if X = 1 and Y = 1 0 Otherwise 0 Otherwise 0 Otherwise AND 01 10 OR o o 1 1 o 1 o 1 1 o 1 1 1 NOT 01 10
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Boolean Switching Algebras A Boolean Switching Algebra is one which deals only with two-valued variables. Boole's general theory covers algebras which deal with n-valued variables.
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Closure Property Closure: When you combine any two elements of the set, the result is also included in the set. A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set. If an element outside the set is produced, then the operation is not closed. For all x, y e B (Boolean Algebra) Here the result ofthe two operations , ie x+y and x.y is either O or 1, so the closure property holds good in Boolean algebra
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I...uaw Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result,
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Associative Law These laws state that
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Identity Law In Boolean algebra there exist identity elements 0 (additive element) and 1 (multiplicative identity) such that
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Inverse Law There exists an inverse such that X + X' = I He will marry or he will not marry : TRUE He will marry and he will not marry : FALSE
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.......................................erinciple of Duality All the laws presented till now have two forms for each law. The duality principle helps in simplifying the proof of both forms everywhere. It states that: If a theorem/property holds good in Boolean algebra , then by i. ii. iii. Interchanging 0 and 1 Interchanging + and Keeping the form of variables as such also holds true. The result obtained by doing so is called dual of the theorem/property
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Idempotent Law This law states that x+x=x x.x = x Idempotence describes the property of operations in mathematics and computer science that yield the same result after the operation is applied multiple times.
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Distributive Law This law states that These DO NOT hold good in ordinary algebra but are true in Boolean algebra.
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Absorption Law This law states that x +x,y = = x x+x',y = = x,((x+y)+z) = z du I dual (x,y)+(x',z)+(y,z) = (x,y)+(x',z)
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Unique Complement theorem If X + Y = 1 and XX = O then X = Y' If RAM's passing OR Krishna' passing is TRUE And RAMA's passing AND Krishna's passing is FALSE Then Either RAMA failed OR Krishna has failed but not both. i.e. If RAMA failed then Krishna has passed or vice-versa.
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Boundedness Law RAMA has passed OR The earth is round is always true because the second statement is always TRUE RAMA has passed AND The earth is flat is always false because the second statement is always TRUE
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Elimination Law
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Consensus theorem or dual form as below Question: Prove the above theorem
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Involution Law This law states that Please don't avoid coming to my party You mean ,I must come to the party ?
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..............................................De Morgan's Law This is a very interesting and important law which is useful in designing logic networks. This law states that (x+y)' = x',y' (x,y)' = x' + y'
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Exercises Using the laws of Boolean algebra, verify the following equations algebraically. x.y' + y'.z = x.y'.z + x.y'z' + x'.y'.z xy + yz + yz' = Y xy+yz + y 'z =xy+z x.y + x'.z + y.z = x.y + x'.z x' y' z' + x' y' z + x'yz + x'yz' + xy'z' + xy'z
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Exercises contd. , , Using the laws of Boolean algebra, verify the following equations algebraically. x' y' z' + x' y' z + xy'z = y'(x'+z) xy' + x' y = (xy + x' y ')' (x+y+z)(xyz)' = xy' + yz' + zx' xy' + yz' + zx' = x' y + y 'z + z' x (a'+b').(a'+b).(a+b') = a' .b'
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VENN DIAGRAMS Boolean Logic Expressions can be expressed figuratively using Venn Diagrams F A.B.C c
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Algebraic Manipulation Minterms and Maxterms A minterm is the product of N distinct literals where each literal occurs exactly once. A maxterm is the sum of N distinct literals where each literal occurs exactly once.
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For a two-variable expression, the minterms and maxterms are as follows x 0 o 1 1 Y o 1 o 1 Minterm X'.Y' X'.Y X.Y' X.Y Maxterm X+Y X+Y' X'+Y For a three-variable expression, the minterms and maxterms are as follows O O o o I 1 1 1 O o I 1 O o I 1 Z o 1 o 1 o 1 o 1 Minterm X'.Y'.Z' X'.Y'.Z X'.Y.Z' X'.Y.Z X.Y'.Z' X.Y'.Z X.Y.Z' X.Y.Z Maxterm X+Y+Z X+Y+Z' X+Y'+Z X+Y'+Z' X'+Y+Z X'+Y+Z' X'+Y'+Z X'+Y'+Z' A minterm is the product of N distinct literals where each literal occurs exactly once A maxterm is the sum of N distinct literals where each literal occurs exactly once SoP To derive the Sum of Products form from a truth table, OR together all of the minterms which give a value of 1. Pos To derive the Product of Sums form from a truth table, AND together all of the maxterms which give a value of 0.
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Sum of Products (SOP) o o 1 1 o 1 o 11 o 1 X.Y Minterm m3 symbol ml m2 SOP is = X.Y' + XX Product of Sum (POS) A minterm is the product of N distinct literals where each literal occurs exactly once A maxterm is the sum of N distinct literals where each literal occurs exactly once o o 1 1 o 1 o 11 1 o 1 Maxterm X+Y X+Y' X'+Y symbol MO Ml IV13 SoP To derive the Sum of Products form from a truth table, OR together all of the minterms which give a value of 1. Pos To derive the Product of Sums form from a truth table, AND together all of the maxterms which give a value of 0. SOP is = (X+Y')
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Exercises (SOP)
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ياس د ح ه ح ت س
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Exercises (POS) Answer to previous problem : F = x'y'z + x'yz' + xy'z' + xyz Write the POS form of a Boolean function F, which is represented by the following truth table
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ه مه ة لى طر لى ت S نم Z قى
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Answer to previous problem : F = Conversion Conversion between SOP and POS is done by application of DeMorgans Laws. Simplification Simplification of expressions can be performed with Boolean algebra. Example: Show that (X.Y' + + Y).z = X.Z + Y.Z = (X.Y' + Z.X + Y'.Z).Z = X.Y'Z + Z.X + Y'.Z = Z.(X.Y' + X + Y') = Z.(X+Y') QED
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Other Logic Operations Boolean Functions Operator Symbol Fl = xy F2 = xy' F4 = x' y FIO = y' FO = 0 F3 = x F6 = xy'+x'y F8 = (x+y)' F9 = xy +x'y' x/y Y/x XCy [email protected] Y' Name Null AND Inhibition Transfer Inhibition Transfer Exclusive-OR OR NOR Equivalence Complement Comments Binary constant 0 x and y X but not y x y but not x xor y but not both x or y Not-OR x equals y Not y
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Boolean Functions F14= (xy)' F15=1 Other Logic Operations Fll = x + y F12 = x' F13 + y Operator Symbol X cy x' xty Name Implication Complement Implication NAND Identity Comments If y then x Not x If x then y Not-AND Binary constant

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