DM2020 boolean algebra
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This document discusses Boolean algebra and logic gates. It begins with an introduction to binary logic and Boolean variables that can take on values of 0 or 1. It describes logical operators like AND, OR, and NOT. Boolean algebra provides a mathematical system for specifying and transforming logic functions. The document provides examples of Boolean functions and logic gates. It discusses topics like Boolean variables and values, Boolean functions, logical operators, Boolean arithmetic, theorems, and algebraic proofs. George Boole is credited with developing Boolean algebra. Truth tables and Karnaugh maps are shown as ways to analyze Boolean functions.
![Logicly [logicly.ly]
The application “Logicly” can be used to simulate the circuit as
shown:
You can take a free trial and then either download it on your
computer to use it or use it online.
27](https://image.slidesharecdn.com/dm2020booleanalgebra-200411114557/85/DM2020-boolean-algebra-27-320.jpg)
DM2020 boolean algebra
- 2. Binary Logic and Gates • Binary variables take on one of two values. • Logical operators operate on binary values and binary variables. • Basic logical operators are the logic functions AND, OR and NOT. • Logic gates implement logic functions. • Boolean Algebra: a useful mathematical system for specifying and transforming logic functions. • We study Boolean algebra as a foundation for designing and analyzing digital systems! Boolean Algebra and Logic Gates 2
- 3. Boolean Variables and Values A Boolean variable, usually denoted in common letters, as a, b,..x. y etc can take only two values 0 and 1. So we can say x = 0 or x = 1 and y = 0 or y = 1 but these are the ONLY values the Boolean variables x and y can take. 3
- 4. Boolean functions • A Boolean function F(x, y) takes Boolean variables and outputs a Boolean value. • For instance, we can have F(x, y) = x + x.y where x and y are Boolean variables. • Again, F(x, y) = xy’ as taken from the Unit notes 4
- 5. Logical Operators • In the last slide, we had the Boolean function F(x, y) = x + x.y • Does the function expression involve operators? 5
- 6. Logical Operators F(x, y) = x + x.y 6 What are these operators?
- 7. Logical Operators • Logical operators operate on binary values and binary variables. • For instance 0.1 = 0 and 1 + 1 = 1 so we have to learn a new arithmetic! 7
- 8. George Boole (1815-1864) 8 George Boole was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland.
- 9. Boolean Arithmetic 4–9 OR AND 0 + 0 = 0 0 * 0 = 0 0 + 1 = 1 0 * 1 = 0 1 + 0 = 1 1 * 0 = 0 1 + 1 = 1 1 * 1 = 1 NEGATION 0’ = 1 1’ = 0
- 10. Boolean Single Value Theorems 4–10 x 1 x + 1 0 1 1 1 1 1 We can prove the above assertions using truth tables as follows: How can you interpret this truth table?
- 11. 11 Absorption Law Use a truth table to show that for two Boolean variables x, y and z: x + xy = x x y xy x + xy 0 0 0 1 1 0 1 1
- 12. 12 Absorption Law Use a truth table to show that for two Boolean variables x, y and z: x + xy = x x y xy x + xy 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 1
- 13. Boolean Expression Calculator 13 https://www.dcode.fr/boolean-expressions-calculator
- 15. Boolean Algebraic Proofs • Our primary reason for doing proofs is to learn: – Careful and efficient use of the identities and theorems of Boolean algebra, and – How to choose the appropriate identity or theorem to apply to make forward progress, irrespective of the application. Boolean Algebra and Logic Gates 15
- 16. Boolean Algebra 16 Boolean algebra refers to symbolic manipulation of expressions made up of boolean variables and boolean operators. The familiar identity, commutative, distributive, and associative axioms from algebra define the axioms of Boolean algebra, along with the two complementary axioms.
- 17. Use Boolean algebra to prove that: x + x · y = x Proof Steps Justification x + x · y = x · 1 + x · y Identity element: x · 1 = x = x · ( 1 + y) Distributive = x · 1 1 + y = 1 = x Identity element Boolean Algebra and Logic Gates 17
- 18. Boolean Algebra 18 In addition, we also have the following;
- 19. Use Boolean algebra to show that (x + y) (x + z) = x + yz 19 Proof Steps Justification (x + y)(x + z) = x x + xz + yx + yz Expanding = x + xz + yx + yz Idempotent Law of Multiplication = x + x (z + y) + yz Distributive Law = x + yz Absorption Law
- 20. Use Boolean algebra to show that (x + y) (x + z) = x + yz 20 Proof Steps Justification (x + y)(x + z) = x + yz Absorption Law
- 21. Is it still true that: (x + y)(x + z) = x(x + z) + y (x + z) 21 x y z x + y x + z (x +y)(x + z) x(x + z) y(x + z) x(x+z)+y(x+z) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
- 22. Is it still true that: (x + y)(x + z) = x(x + z) + y (x + z) 22 x y z x + y x + z (x +y)(x + z) x(x + z) y(x + z) x(x+z)+y(x+z) 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
- 23. Boolean Functions 23 A Boolean function accepts binary inputs and outputs a single binary value. A Boolean function is a mathematical function that maps arguments to a value where the allowable values of range (the function arguments) and domain (the function value) are just one of two values, 0 or 1. We usually denote Boolean functions as F(x, y) where x and y are Boolean variables. We use truth tables to evaluate the functions. https://introcs.cs.princeton.edu/java/71boolean/
- 24. Boolean function • We can define a Boolean function: F(x, y) = (x AND y) OR (NOT x AND NOT y) • We write: F(x, y, z) = xy + x’y’ • and then write a “truth table” for it: 24 x y x’ y’ xy x’y’ F(x, y ,z) = xy + x’y’) 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 • Finally, use gates to implement the function: F(x, y) = xy + x’y • We use two AND gates and one OR gate.
- 25. Boolean function • Another example: F(x, y, z) = (x AND y AND (NOT z)) OR (x AND (NOT y) AND z) OR ((NOT x) AND Y AND Z) OR xyz • We write this as: F(x, y, z) = xyz’ + xy’z + x’yz + xyz 25 • Can this function be simplified? • We can use the Boolean Expression Calculator to help with this…
- 26. Boolean Expression Calculator 26 https://www.dcode.fr/boolean-expressions-calculator
- 27. Logicly [logicly.ly] The application “Logicly” can be used to simulate the circuit as shown: You can take a free trial and then either download it on your computer to use it or use it online. 27
- 28. Building the circuit in logicly 28
- 29. The gate circuit and truth table 29
- 30. Logic Gate Circuit 30 What practical use would this circuit have? Suggestions…
- 31. Minterms A minterm, denoted by mi, where 0 ≤ I < 2n, is a product (AND) of the n variables in which each variable is complemented if the value assigned to it is 0, and uncomplemented if it is 1. We speak of the 1-minterms = minterms for which the function F = 1 and 0-minterms, the minterms for which the function F = 0. Any Boolean function can be expressed as a sum (OR) of its 1- minterms. We write F(x, y z) = ∑(3, 5, 6, 7)
- 32. Maxterms 32
- 33. Boolean Function F(x, y, z) = x + yz 33 The truth table can show us all the outputs of F(x, y, z) for various values of the inputs:. x y z yz x + yz 0 0 0 0 m0 = 0 0 0 1 0 m1 = 0 0 1 0 0 m2 = 0 0 1 1 1 m3 = 1 1 0 0 0 m4 = 1 1 0 1 0 m5 = 1 1 1 0 0 m6 = 1 1 1 1 1 m7 = 1 So here we have F(x, y, z) = x’yz + xy’z’ + xy’z + xyz’ +xyz
- 34. Boolean Expression Calculator 34 F(x, y, z) = x’yz + xy’z’ + xy’z + xyz’ +xyz Spot the error?
- 36. Boolean Expression Calculator 36 F(x, y, z) = x’yz + xy’z’ + xy’z + xyz’ +xyz = x + yz
- 37. 37 F(x, y, z) = x’yz + xy’z’ + xy’z + xyz’ +xyz = x + yz How many gates does each of the expressions use?
- 38. Recall: (x + y) (x + z) = x + yz and now we have x + yz = x’yz + xy’z’ + xy’z + xyz’ +xyz Is there a way to prove all this algebraically… 38 Proof Steps Justification x’yz + xy’z’ + xy’z + xyz’ +xyz = x’yz + xy’z’ + xy’z + xyz’ +xyz + xyz Idempotent = (x’ + x) yz + xy (z + z’) + xy’(z + z’) Distributive = yz + xy + xy’ Complement law = x(y + y’) + yz Distributive Law = x + yz Absorption law
- 39. Show that: x’yz + xy’z + xyz’ +xyz = xy + yz + + yz 39 Proof Steps Justification x’yz + xy’z + xyz’ +xyz = x’yz + xy’z + xyz’ +xyz +xyz + xyz Idempotent = xy(z + z’) + xz (y + y’) + yz (x + x’) Distributive = xy + yz + xz Complement law
- 40. We have seen that (x + y) (x + z) = x + yz 40 (x + y) (x + z) is a Product of Sums (POS) expression snd x + yz is a Sum of Products (SOP) So F(x, y z) = (x + y)(x + z) = x + yz Ie. The same function can be represented in SOP or POS format.
- 41. Express the function F(x, y, z) = x + yz as a POS expression. 41 Proof Steps Justification x + yz = x + x (z + y) + yz = x + xz + xy + yz = xx + xz + xy + yz = (x + y)(x + z) as per required.
- 42. Definitions 42 The boolean expression that we construct is known as the sum-of-products representation is also known as the disjunctive normal form of the function. A boolean expression consisting entirely either of minterm or maxterm is called canonical expression.
- 43. 43 Express the Boolean function F(a, b, c) = (b + ac’)’(ab’ + c) in Sum-of-Products form using Boolean algebra laws and theorems. Express in the minimal expression. F(a, b, c) = (b + ac’)’(ab’ + c) = (b’)(ac’)’(ab’ + c) De Morgan’s Law = b’(a’ + c)(ab’ + c) De Morgan’s law = (a’b’ + b’c)(ab’ + c) Distributive Law = a’b’ab’ + a’b’c + b’cab’ + b’cc = a’b’c + ab’c + b’c = b’c(a’ + a + 1) = b’c
- 44. 44 We had earlier expressed the Boolean function F(a, b, c) = (b + ac’)’(ab’ + c) in Sum-of-Products form using Boolean algebra laws and theorems. Express this in canonical form. F(a, b, c) = (b + ac’)’(ab’ + c) a b c b’ c’ ac’ b + ac’ (b + ac’)’ ab’ ab’ + c F(a, b, c) 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 F(a, b, c) = a’b’c + ab’c
- 45. Canonical Form • Express in canonical form…F(X, Y, Z) = X Y + Y ‘Z X Y Z X Y Y’ Y ‘Z F = X Y + Y ‘Z 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 1
- 46. 46
- 47. Claude Shannon 47 Shannon's work became the foundation of digital circuit design, as it became widely known in the electrical engineering community during and after World War II. The theoretical rigor of Shannon's work superseded the ad hoc methods that had prevailed previously..
- 48. Boolean Algebra Simplification Boolean Algebra and Logic Gates 48 11 1 1 11 1 0 11 0 1 11 0 0 00 1 1 00 1 0 10 0 1 00 0 0 X Y Z F(x, y, z) From the minterms expansion, we have: F(x, y, z) = x’y’z + xy’z’ +xy’z + xyz’ + xyz = (x’ + x)y’z + xz’(y’ + y) + xyz = y’z + xz’y’ + xz’y + xyz = y’z + x(yz + yz’ + z’y’) = y’z + x(yz + yz’ + (yz)’) = y’z + x(1 + yz’) = x + y’z We can use the Boolean algebra laws to simplify a function. What is the formula for F9x, y, z) from the following?
- 49. From the Boolean expression to the logic gate diagram… Boolean Algebra and Logic Gates 49 11 1 1 11 1 0 11 0 1 11 0 0 00 1 1 00 1 0 10 0 1 00 0 0 X Y Z F(x, y, z) F(x, y, z) = x’y’z + xy’z’ +xy’z + xyz’ + xyz = (x’ + x)y’z + xz’(y’ + y) + xyz = y’z + xz’y’ + xz’y + xyz = y’z + x(yz + yz’ + z’y’) = y’z + x(yz + yz’ + (yz)’) = y’z + x(1 + yz’) = x + y’z We can use the Boolean algebra laws to simplify a function. What is the formula for F9x, y, z) from the following? X Y F Z
- 50. Logic gates… 50