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Boolean Algebra and Computer Logic Mathematical Structures for Computer Science Chapter 7.1 – 7.2 Copyright © 2006 W.H. Freeman & Co.MSCS Slides Boolean Logic and Computer Logic

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Review: Boolean algebra Definition A Boolean algebra consists of a set of elements B two binary operations and one unary operation two distinct elements 0 and 1 such that the following properties hold for all x, y, z B (Idempotency can be derived from these properties) x + x = x x * x = x Properties 1. x + y = y + x (commutative properties) 2. (x + y) + z = x + (y + z) (associative properties) 3. x + (y * z) = (x + y) * (x + z) (distributive properties) 4. x + 0 = x (identity properties) 5. x + x = 1 (complement properties) And 1. x * y = y * x (commutative properties) 2. (x* y) * z = x * (y * z) (associative properties) 3. x * (y + z) = (x * y) + (x * z) (distributive properties) 4. x * 1 = x (identity properties) 5. x * x = 0 (complement properties) Section 7.2Logic Networks1

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Example Let B = {0, 1} and define + by x + y = max(x, y), * by x * y = min(x, y), and ′ as follows: 0′ = 1, 1′ = 0. It is possible to verify all of the properties in the definition of a Boolean algebra using brute force. For example for property 2, associativity of *: (0*0)*0 = 0*(0*0) = 0 (0*0)*1 = 0*(0*1) = 0 (0*1)*0 = 0*(1*0) = 0... (1*1)*1 = 1*(1*1) = 1 Property 4, identity for addition: x + 0 = x For x = 0, 0 + 0 = 0 For x = 1, 1 + 0 = 1 Although it is tedious you can use this approach to show that [{0, 1}, max, min, (0′ = 1, 1′ = 0), 0, 1] satisfies the requirements of a Boolean algebra. Section 7.2Logic Networks2

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Tautological Equivalences in Propositional Logic Example: Let be disjunction (or) and be conjunction (and); A′ is the negation of A, 0 is a contradiction, 1 is a tautology. then A B = B A, A B = B A (commutative), A 0 = A, A 1 = A (identity property) A A′ = 1, A A′ = 0 (complement property) Associative and distributive properties can also be shown Thus Propositional Logic has all the properties of a Boolean algebra Equivalences Among the Subsets of a Universal Set S Example: Let and be set union and inter- section, A′ is the complement of a set, is the empty set, S is the universal set. Then A B = B A, A B = B A (commutative), A = A, A S = A (identity property) A A′ = 1, A A′ = (complement property) Associative & distributive properties can also be shown Set identities have the properties of a Boolean algebra. Section 7.2Logic Networks3 Review

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Examples of Boolean Algebras Propositional Logic [B = {statements}, + and * = and , the unary operator contradicts (changes truth value of) a statement, and contradiction and tautology are equivalent of 0 and 1]. (0 is a statement that is always false, 1 is a statement that is always true.) Subsets of some set S also are the elements that form a Boolean algebra under the operations , , and set complementation, with 0/1 elements being and S. Section 7.2Logic Networks4

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Section 7.2Logic Networks5 Combinational Networks Basic logic elements Imagine that the electrical voltages carried along wires fall into one of two ranges, high or low, which we shall represent by 1 and 0, respectively. Voltage fluctuations within these ranges are ignored, so we are forcing a binary mask on an analog phenomenon. Also suppose that switches can be wired so that a signal of 1 causes the switch to be closed and a signal of 0 causes the switch to be open as seen in figure below. Combine two such switches, controlled by lines x 1 and x 2, in parallel. The figure below illustrates the various cases. Fig. 7.6

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Combinational Networks Table 7.2 summarizes the behavior of the circuit in figure 7.6 x 1 x 2 Output 1 1 1 1 0 1 0 1 1 0 0 0 If you interpret 1 as True and 0 as False (as most programming languages do) then this is the truth table for disjunction (logical or). Section 7.2Logic Networks6

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Section 7.2Logic Networks7 Basic Logic Elements The OR gate, Figure (a), behaves like the Boolean operation +. (x + 0 = x, x + x = x) The AND gate, Figure (b), represents the Boolean operation *. (x * 1 = x, x * x = x) Figure (c) shows an inverter, corresponding to the unary Boolean operation. Because of the associativity property for and, the OR and AND gates can have more than two inputs.

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Section 7.2Logic Networks8 Boolean Expressions DEFINITION: BOOLEAN EXPRESSION A Boolean expression in n variables, x 1, x 2,..., x n, is any finite string of symbols formed by applying the following rules: 1. x 1, x 2,..., x n are Boolean expressions. 2. If P and Q are Boolean expressions, so are (P + Q), (P Q), and (P). x 3, (x 1 + x 2 )x 3, (x 1 x 3 + x 4 )x 2, and (x 1 x 2 )x 1 are all Boolean expressions. In propositional logic, the logical connectives , , and are instances of the operations of a Boolean algebra.

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Section 7.2Logic Networks9 Truth Functions DEFINITION: TRUTH FUNCTION A truth function is a function f such that f:{0,1} n {0,1} for some integer n 1. The notation {0,1} n denotes the set of all n-tuples of 0s and 1s. A truth function thus associates a value of 0 or 1 with each such n-tuple. Example: The Boolean expression x 1 x 2 + x 3 defines the truth function given in Table 7.3 below:

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Section 7.2Logic Networks10 Networks and Expressions By combining AND gates, OR gates, and inverters, we can construct a logic network representing a given Boolean expression that produces the same truth function as that expression. Example: Logic network for Boolean expression x 1 x 2 + x 3 is shown in the figure below: Example: Logic network for Boolean expression (x 1 x 2 + x 3 ) + x 3 is shown in the figure below:

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