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Introduction to Boolean Algebra & Logic Gates: Fundamentals and Applications

Explore the basic definitions, theorems, and properties of Boolean algebra, logical operations, digital logic gates, and canonical forms. Understand Boolean functions, truth tables, DeMorgan's Theorem, and the conversion between canonical forms effectively. Ideal for students and professionals.

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Introduction to Boolean Algebra & Logic Gates: Fundamentals and Applications

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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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