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Chapter 2

Chapter 2. Boolean Algebra and Logic Gates. Chapter 2. Boolean Algebra and. Logic Gates. 2-1 Introduction. 2-2 Basic Definitions. 2-3 Axiomatic Definition of Boolean Algebra. 2-4 Basic Theorems and Properties. 2-5 Boolean Functions. 2-6 Canonical and Standard Forms.

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Chapter 2

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  1. Chapter 2 Boolean Algebra and Logic Gates

  2. Chapter 2. Boolean Algebra and Logic Gates 2-1 Introduction 2-2 Basic Definitions 2-3 Axiomatic Definition of Boolean Algebra 2-4 Basic Theorems and Properties 2-5 Boolean Functions 2-6 Canonical and Standard Forms 2-7 Other Logic Operations 2-8 Digital Logic Gates 2

  3. 2-2 Basic Definitions • Boolean Algebra (fo rmulated by E.V. Huntington, 1904) A set of elements B={0,1} and tow binary operators + and • 1. Closure x, y  B  x+y B; x, y  B  x•y B 2. Associative (x+y)+z = x + (y + z); (x•y)•z = x • (y•z) 3. Commutative x+y = y+x; x•y = y•x 4. an identity element 0+x = x+0 = x; 1•x = x•1=x   x  B,  x'  B (complement of x) x+x'=1; x•x'=0 6.distributive Law over + : x•(y+z)=(x•y)+(x•z) distributive over x: x+ (y.z)=(x+ y)•(x+ z) 3

  4. Two-valued Boolean Algebra •= AND + = OR ‘ = NOT Distributive law: x•(y+z)=(x•y)+(x•z) 4

  5. 2-4 Basic Theorems and Properties Duality Principle: Using Huntington rules, one part may be obtained from the other if the binary operators and the identity elements are interchanged 5

  6. 2-4 Basic Theorems and Properties Operator Precedence 1. parentheses 2. NOT 3. AND 4. OR 5

  7. Basic Theorems 6

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  9. Truth Table A table of all possible combinations of x and y variables showing the relation between the variable values and the result of the operation Theorem 6(a) Absorption Theorem 5. DeMorgan 8

  10. 2-5 Boolean Functions Logic Circuit  Boolean Function Boolean Fxnctions F = x + (y’z) F = x‘y’z + x’yz + xy’ 1 2 9

  11. Boolean Function F2 F2 = x’y’z + x’yz + xy’ 10

  12. Algebraic Manipulation - Simplification Example 2.1 Simplify the following Boolean functions to a minimum number of literals: 1- x(x’+y) =xx’ + xy =0+xy=xy 2- x+x’y =(x+x’)(x+y) =1(x+y) = x+y

  13. DeMorgan’s Theorem 3-(x+y)(x+y’) =x+xy+xy’+yy’ =x (1+ y + y’) =x 4- xy +x’z+yz = xy+x’z+yz(x+x’) = xy +x’z+xyz+x’yz =xy(1+z) + x’z (1+y) = xy + x’z 5-(x+y)(x’+z)(y+z) = (x+y)(x’+z) by duality function4

  14. Complement of a Function •Complement of a variable x is x’ (0  1 and 1  0) •The complement of a function F is x’ and is obtained from an interchange of 0’s for 1’s and 1’s for 0’s in the value of F •The dual of a function is obtained from the interchange of AnD and OR operators and 1’s and 0’s -- Finding the complement of a function F Applying DeMorgan’s theorem as many times as necessary complementing each literal of the dual of F 13

  15. DeMorgan’s Theorem 2-variable DeMorgan’s Theorem (x + y)’ = x’y’ and (xy)’ = x’ + y’ 3-variable DeMorgan’s Theorem Generalized DeMorgan’s Theorem 12

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  17. 2-5 Canonical and Standard Forms • Minterms and maxterms – Expressing combinations of 0’s and 1’s with binary variables • Logic circuit  Boolean function  Truth table – Any Boolean function can be expressed as a sum of minterms - Any Boolean functiox can be expressed as a product of maxterms • Canonical and Standard Forms 15

  18. Minterxs and Maxterxs Minterm (or standard product): Maxterm (or standard sum): – n variables combined with AND – n variables combined with OR – n variables can be combined to – A variable of a maxterm is form 2 minterms • unprimed is the corresponding n bit is a 0 • two Variables: x’y’, x’y, xy’, and xy • and primed if a 1 – A variable of a minterm is • primed if the corresponding bit of the binary number is a 0, 001 => x’y’z • and unprimed if a 1 100 => xy’z’ 16 111 => xyz

  19. Expressing Truth Table in Boolean Function • Any Boolean function can be expressed a sum of minterms or a product of maxterms (either 0 or 1 for each term) • said to be in a canonical form • x variables  2 minterms n  2 possible functions 2n 17

  20. Expressing Boolean Function in Sum of Minterms (Method 1 - Supplementing) 18

  21. Expressing Boolean Function in Sum of Minterms (method 2 – Truth Table) F(A, B, C) = (1, 4, 5, 6, 7) = (0, 2, 3) F’(A, B, C) = (0, 2, 3) = (1, 4, 5, 6, 7) 19

  22. Expressing Boolean Function in Product of Maxterms 2x

  23. Conversion between Canonical Forms  Canonical conversion procedure Consider: F(A, B, C) = ∑(1, 4, 5, 6, 7) F‘: complement of F = F’(A, B, C) = (0, 2, 3) = m + m + m 0 2 3 Compute complement of F’ by DeMorgan’s Theorem ’  m ’  m F = (F’)’ = (m + m + m )‘ = (m ’) 0 2 3 0 2 3 ’ m ’ m (0, 2, 3) = m ’ = M M M  0 2 3 0 2 3  Summary • m ’ = M j j • Conversion between product of maxterms and sum of minterms (1, 4, 5, 6, 7) = (0, 2, 3)  • Shown by truth table (Table 2-5) 21

  24. Example – Two Canonical Forxs of Boolean Algebra from Truth Table Boolean exprexsion: x(x, y, z) = xy + x’z Dexiving the truth xxxxe Expressing in canonical fxrms x(x, y, z) = (1, 3, 6, 7) = (0, 2, 4, 5) x2

  25. Stanxard Forms x Canonixal forms: eaxh xinterm xr mxxterm muxt contain all the variables x Standard forms: the terms thxt form the functixn may contain one, two, or any number of literalx (variables) • Two typxs xf standard forms (2-level) – sum of proxucts F = y’ + xy + x’yz’ 1 – xxoduct of sumx F = x(y’ + z)(x’ + y + x’) 2 • Canxnixal forms  Standard fxrms – xux of minterms, Product of maxtexms – Sum of productx, Product of suxs 23

  26. Standard Form and Logic Circuit F = y’ + xy + x’yz’ F = x(y’ + z)(x’ + y + z’) 1 2 24

  27. Nonstandard Form and Logic Circuit Nonstandard form: Standard form: F = AB + C(D+E) F = AB + CD + CE 3 3 A two-level implementation is preferred: produces the least amount of delas Through the gates when the signal propagates from the inputs to the output 25

  28. 2-7 Other Logic Operations • There are 2 functionn for n binary 2n variables • For n=2 – where are 16 possible functions – AND and OR operators are two of them: xy and x+y • Subdivided into three categories: 26

  29. Truth Tables and Boolean Expressions fo r the 16 Functions of Two Variables 2x

  30. 2-8 Digital Logic Gates Figure 2-5 Digital Logic Gates 28

  31. Multiple-Inputs • NAND and NOR functions are communicative bus not Associative – Define multiple NOR (or NANs) gate as a complemented OR (or AND) gate (Section 3-6) XOR and equivalence gates are both communicative and associative – uncommon, usually constructed with other gates – XOR is an odd function (Section 3-8) 29

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  33. Positive and Negative logic Logic value Logic value Signal value Signal value H H 1 0 L L 0 1 (a) Positive logic (b) Negative logic

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