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Venn Diagram – the visual aid in verifying theorems and properties 1 E

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Venn Diagram in Boolean algebra Represent the universe B = {0, 1} by a square. {1} using shaded area Represent a Boolean variable x by a circle. Area inside the circle -> x = 1; Area outside the circle -> x = 0; 2 x (a) Constant 1(b) Constant 0 (c) Variable x (d) x

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Venn Diagram – for two or more Boolean variables Represent x, y by drawing two overlapping circles AND operation x ∙ y -> shade overlapping area of both circles. -> also referred to as the intersection of x and y. OR operation x + y -> shade total area within both circles -> also called the union of x and y 3 xy z x xyxy (e)(f) (g) (h) xy xy+ xyz+ xy y

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App: Verifying the equivalence of two expressions 4 xy z xy z xy z xy z xy z xy z x xy xy x+z xyz+ (a) (d) (c) (f) xz yz+ (b) (e) Verification of distributive property x ∙ (y + z) = x ∙ y + x ∙ z

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Another verification example 5 xy z yx z xy z xy y z z xy z xy xy z z y z xy z x

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Figure 2.15. A function to be synthesized. 2.6 Synthesis using AND, OR, NOT gates Can express the required behavior using a truth table 6

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Procedures for designing a logic circuit Create a product term for each valuation whose output function f is 1. –Product term: all variables are ANDed. Take a logic sum (OR) of these product terms to realize f. 7

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f (a) Canonical sum-of-products f (b) Minimal-cost realization x 2 x 1 x 1 x 2 Figure 2.16. Two implementations of a function in Figure 2.15. 8

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Summary 9

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Minterms and Sum-of-products (SOP) Minterms: a product term in which each of the n variables for a function appear once –Variables may appear in either un-complemented or complemented form, –Use m i to denote the minterm for the row number i. Sum-of-products Form: a logic expression consisting of product (AND) terms that are summed (ORed) –Canonical SOP: each term is a minterm 10

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Figure 2.17 Three-variable minterms and maxterms. 11

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Figure 2.18. A three-variable function. 12

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Maxterms and Product-of-Sums (POS) Maxterms: complements of minterms –By applying the principle of duality, if we could synthesize a function f by considering the rows for which f = 1, it should also be possible to synthesize f by considering the rows where f = 0 Product-of-sums Form: a logic expression consisting of sum (OR) terms that are the factors of a logical product (AND) –Canonical POS: each term is maxterm 13

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Figure 2.17 Three-variable minterms and maxterms. 14

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An example 15

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Figure 2.18. A three-variable function. 16

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Figure 2.19. Two realizations of a function in Figure 2.18. f (a) A minimal sum-of-products realization x 1 x 2 x 3 Cost of a logic circuit is –the total number of gates plus –the total number of inputs to all gates in the circuit. 17

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Figure 2.19. Two realizations of a function in Figure 2.18. f (a) A minimal sum-of-products realization f (b) A minimal product-of-sums realization x 1 x 2 x 3 x 2 x 1 x 3 18 Cost = 13

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Example 2.3 19

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Example 2.4 20

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Discussion (1) 21 Complemented entry -> 0 uncomplement entry -> 1

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Discussion (2) 22 Complemented entry -> 1 uncomplement entry -> 0

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Venn Diagram for Boolean algebra Basic requirement for legal Venn diagram –Must be able to represent all minterms of a Boolean function 23 Two variables x1x1 x2x2 m1m1 m0m0 m3m3 m2m2 Three variables m0m0 m7m7 x1x1 x2x2 x3x3 m2m2 m6m6 m3m3 m5m5 m4m4 m1m1

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Venn Diagram for Boolean algebra Basic requirement for legal Venn diagram –Must be able to represent all minterms of a Boolean function 24 Two variables x1x1 x2x2 m1m1 m0m0 m3m3 m2m2 Three variables m0m0 m7?m7? x1x1 x2x2 x3x3 m2m2 m6m6 m3m3 m5?m5? m4m4 m1m1

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