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Boolean Algebra Boolean Algebra: Structure Instudying propositional logic (true, false), we learned about certain operators, that can be viewed as functions f:{T, F}2→{T, F}. Forexample, (T, T) = T, (T, F) = T, (F, T) = T, and (F, F) = F. From there, we learned about certain fundamental properties of wff's: Boolean Algebra: Structure ABBA ABBA commutative laws (A B) C A (B C) (A B) C A (B C) associative laws A (B C) (A B) (A C) A (B C) (A B) (A C) distributive laws A F A A T A identity laws A A' T A A' F complement laws Boolean Algebra: Structure We also studied sets, and within Set Theory we discovered that given sets A, B and C within a universe U: Boolean Algebra: Structure AB=BA AB=BA commutative laws (A B) C = A (B C) (A B) C = A (B C) associative laws A (B C) = (A B) (A C) A (B C) = (A B) (A C) distributive laws AF=A AU=A identity laws A A' = U A A' = complement laws Boolean Algebra: Structure One of the beauties of mathematics is that there is a lot of consistency and many recurring patterns. Many different models satisfy the same (or similar) properties. Systems like the ones we have just seen are examples of a model called a Boolean algebra. Boolean Algebra: Structure A Boolean algebra is a set B on which are defined two binary operations: + and and one unary operation ' and in which there are two distinct elements 0 (additive identity) and 1 (multiplicative identity) such that the following properties hold for all x, y, z B: Boolean Algebra: Structure x+y=y+x xy=yx commutative laws (x + y) + z = x + (y + z) (x y) z = x (y z) associative laws x + (y z) = (x + y) (x + z) x (y + z) = (x y) + (x z) distributive laws x+0=x x1=x identity laws x + x' = 1 x x' = 0 complement laws Boolean Algebra: Structure Example: Consider the set B = {0, 1} and define the operations + and to be x + y = max(x, y) and x y = min(x, y). Let 0' = 1 and 1' = 0. The above forms a Boolean algebra. • commutative: obviously max(x,y) = max(y,x) and the same for min. • associative: intuitively true. • distributive: max(x, min(y,z)) = min(max(x,y), max(x,z)) • identity • complement Boolean Algebra: Structure The idempotent property: x + x = x x + x = (x + x) 1 multiplicative identity = (x + x) (x + x') additive complement = x + (x x') distributive property =x+0 multiplicative complement =x additive identity Boolean Algebra: Structure The Principle of Duality: If some equality holds within a Boolean algebra, then the corresponding equality obtained by substituting • + for , • for +, • 0 for 1 • and 1 for 0 will also be true. Boolean Algebra: Structure Uniqueness: If x + x1 = 1 and x x1 = 0, then x1 = x'. De Morgan: (x + y)' = x' y', (x y)' = x' + y' Absorption: x + (x y) = x, x (x + y) = x Double negation: (x')' = x Boolean Algebra: Structure One more thing: If (B, + , , ') is a Boolean algebra, then |B| = 2n Logic Networks Claude Shannon (1938) Boolean algebras and logic circuits. x=1 x=0 x=1 1 x=0 Logic Networks A B A+B A B A B A A' 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 inverter 0 0 0 0 0 0 OR gate AND gate Logic Networks A B (A + B)' A B (A B)' 1 1 0 1 1 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 NOR gate NAND gate Logic Networks and Truth Tables Consider a truth table involving three variables. There will be 23 = 8 lines. There are 2 23 = 28 = 256 unique truth tables. Each truth table can be represented by logic circuits using the basic gates shown earlier. Logic Networks and Truth Tables Consider the following table: x1 x2 x3 f(x1,x2,x3) 1 1 1 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 Logic Networks and Truth Tables The truth table on the previous slide can be implemented by any number of logic circuits using AND, OR and inverter gates in combination. One such circuit can be built using the sum-of-products notation. Let's go back to that table. Logic Networks and Truth Tables x1 x2 x3 f(x1,x2,x3) 1 1 1 1 x1x2x3 sum of 1 1 0 0 products 1 0 1 1 x1x'2x3 1 0 0 1 x1x'2x'3 0 1 1 0 0 1 0 0 0 0 1 1 x'1x'2x3 0 0 0 0 x1x2x3 + x1x'2x3 + x1x'2x'3 + x'1x'2x3 Logic Circuits and Truth Tables x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 Logic Circuits and Truth Tables But is this the only circuit that corresponds to the earlier truth table? Not by a long shot! Consider the expression x1x3 + x1x'2 + x'2x3 Logic Circuits and Truth Tables x1 x2 x3 x1x3 x1x'2 x'2x3 x 1x 3 + x1x'2+x'2x3 1 1 1 1 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 Logic Circuits and Truth Tables The truth table on the previous slide is identical to the earlier one. Note that it only requires three AND gates leading to the OR gate. Note that each of these AND gates requires only 2 (not 3) inputs. It's simpler. It's cheaper. Minimization There is a technique for finding the minimal circuit for a given truth table. It involves a special map called the Karnaugh Map. Here are the general pictures for Karnaugh Maps corresponding to two- variable, three-variable and four- variable expressions. Karnaugh Maps x1 x'1 x1x2 x1x'2 x'1x'2 x'1x2 x2 1 1 x3 1 1 1 1 x'2 x'3 1 1 1 x1x2 x1x'2 x'1x'2 x'1x2 x3x4 x3x'4 x'3x'4 x'3x4 Karnaugh Maps Note that adjacent squares have only one variable that changes (e.g. – x1 to x'1) Note that Karnaugh Maps "wrap around", top-to-bottom, left-to-right. We look for sub-rectangles (perhaps wrapping around) with both the number of rows and the number of columns being powers of two (1, 2, 4, etc.) Karnaugh Maps Find the smallest number of largest rectangles that cover all the 1's and only the 1's. Each rectangle is a product of variables (or inverses of variables). The simplest expression is the sum of these rectangles. Karnaugh Maps Consider the following map: x1x2 x1x'2 x'1x'2 x'1x2 x3x4 1 1 1 x3x'4 1 1 x'3x'4 1 x'3x4 x1x3 + x'2x3x4 + x1x'2x'4 Karnaugh Maps Consider the following map: x1x2 x1x'2 x'1x'2 x'1x2 x3x4 1 1 1 x3x'4 1 1 x'3x'4 1 x'3x4 x1x3 + x'2x3x4 + x1x'2x'4 Karnaugh Maps Consider the following map: x1x2 x1x'2 x'1x'2 x'1x2 x3x4 1 1 1 x3x'4 x'3x'4 1 x'3x4 1 1 x2x4 + x'1x3x4 + x1x'2x’3x'4 Karnaugh Maps Consider the following map: x1x2 x1x'2 x'1x'2 x'1x2 x3x4 1 1 1 1 x3x'4 1 1 x'3x'4 1 1 x'3x4 1 1 x2 + x3x4

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posted: | 9/30/2012 |

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