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1 Section 10.1 Boolean Functions

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2 Computers & Boolean Algebra Circuits in computers have inputs whose values are either 0 or 1 Mathematician George Boole set forth basic rules of logic, which subsequently were adapted to define basic circuits; these rules form basis of Boolean algebra Operation of a circuit is defined by a Boolean function that specifies the output for each set of inputs

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3 Boolean algebra Boolean algebra provides operations and rules for working with set {0,1} Most common operations are: –complement (NOT) –Boolean sum (OR) –Boolean product (AND) Rules of precedence: 1) complement, 2) product, 3) sum

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4 Notation for Boolean algebra Complement is denoted by bar:0= 1 and1= 0 Boolean sum is denoted by +: 1 + 1 = 1 1 + 0 = 10 + 1 = 10 + 0 = 0 Boolean product is denoted by. Symbol may be omitted 1. 1 = 1 1. 0 = 00. 1 = 00. 0 = 0

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5 Example 1 Find the value of 1. 0 + (0 + 1) = 0 + 1 = 0 + 0 = 0

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6 Boolean algebra & logical operations Boolean algebraic operations correspond to logical operations: –complement = –sum = –product = –0 = F, 1 = T Results of Boolean algebra can be directly translated into results about propositions, and vice-versa

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7 Boolean functions Let B = {0, 1} –a variable x is a Boolean variable if it assumes values only from B –a function from B n = {(x 1, x 2, … x n ) | x i B, 1<=i<= n} to B is a Boolean function of degree n Values of a Boolean function are often displayed in tables resembling truth tables

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8 Boolean expressions Boolean functions can be represented using expressions made up from the variables and Boolean operations Boolean expressions in the variables x 1, x 2, … x n are defined recursively as: –0, 1, x 1, x 2, ….,x n are Boolean expressions; –if E 1 and E 2 are Boolean expressions, then their complements, (E 1 E 2 ) and (E 1 +E 2 ) are Boolean expressions Each Boolean expression represents a Boolean function; values of the function are obtained by substituting 0 and 1 for variables in the expression

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9 Example 2 Find the values of the Boolean function represented by F(x,y,z) = xy + z x y z xy z F(x,y,z) 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1

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10 Properties & Operations on Boolean Functions Boolean functions F and G of n variables are equal if and only if F(b 1, b 2, … b n ) = G(b 1, b 2, … b n ) whenever b 1, b 2, … b n B Boolean expressions that represent the same function are equivalent- e.g. xy, xy+0, xy. 1 Complement of a Boolean function F is the function F where F(x 1,…,x n ) = F(x 1,…,x n )

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11 Boolean sums & products of functions Let F and G be Boolean functions of degree n –Boolean sum F+G is defined by (F+G)(x 1,…,x n ) = F(x 1,…,x n ) + G(x 1,…,x n ) –Boolean product FG is defined by (FG)(x 1,…,x n ) = F(x 1,…,x n )G(x 1,…,x n )

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12 Boolean functions of degree n A Boolean function of degree 2 is a function from a set of 4 elements (pairs of elements from B={0,1}) to B, a set with 2 elements There are 16 different Boolean functions of degree 2: x y f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

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13 Identities of Boolean algebra Identities of Boolean algebra are analogous to logical equivalences These identities are useful in simplifying circuit design Each identity can be proven using a table Identities can be used to prove further identities

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14 Boolean identities Law of the double complement: x = x Idempotent laws: x + x = x and x. x = x Identity laws: x + 0 = x and x. 1 = x Dominance laws: x + 1 = 1 and x. 0 = 0 Commutative laws: x + y = y + x and xy = yx Associative laws: x + (y + z) = (x + y) + z and x(yz) = (xy)z Distributive laws: x + yz = (x + y)(x + z) and x(y + z) = xy + xz DeMorgan’s laws: (xy) = x + y and (x + y) = x. y

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15 Example 3: Proof of DeMorgan’s first law x y x y xy xy x + y 1 1 0 0 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 1 1

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16 Example 4 Prove the absorption law: x(x + y) = x using the identities of Boolean algebra x(x + y) = (x + 0)(x + y)identity law for Boolean sum = x + 0. ydistributive law of Boolean sum over Boolean product = x + y. 0commutative law for Boolean product = x + 0dominance law for Boolean product = x identity law for Boolean sum

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17 Duality Note that most identities come in pairs The relationship between the 2 identities in a pair can be explained using the concept of a dual: the dual of a Boolean expression is obtained by interchanging Boolean sums and Boolean products, and 1s and 0s

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18 Example 5 Find the dual of x(y + 0) Substitute. for + and + for. : x + (y. 0) Substitute 0 for 1: x + (y. 1)

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19 Duality The dual of a Boolean function F represented by a Boolean expression is the function represented by the dual of the expression This dual function F d, does not depend of the particular Boolean expression used to represent F; an identity between functions represented by Boolean expressions remains valid when the duals of both sides of the identity are taken This duality principle is useful for obtaining new identities

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20 Example 6 Construct an identity from the absorption law: x(x + y) = x Taking duals of both sides: –x + (x. y) –x–x Result is x + xy = x, also called the absorption law

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21 Abstract definition of a Boolean algebra Most common way to define a Boolean algebra is to specify properties that operations must satisfy Next slide illustrates such a definition

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22 Definition of a Boolean algebra A Boolean algebra is a set B with two binary operations and , elements 0 and 1 and a unary operation _ such that the following properties hold for all x, y and z in B: Identity laws: x 0 = x and x 1 = x Domination laws: x x = 1 and x x = 0 Associative laws:(x y) z = x (y z) and (x y) z = x (y z) Commutative laws: x y = y x and x y = y x Distributive laws:x (y z) = (x y) (x z) and x (y z) = (x y) (x z)

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23 Section 10.1 Boolean Functions

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