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Venn Diagram – the visual aid in verifying theorems and properties • Provides a graphical illustration of operations and relations in the algebra of sets. – The elements of a set are represented by the area enclosed by a contour. – Given a universe N of integers from 1 to 10; • Even numbers E = { 2, 4, 6, 8, 10 }; • Odd numbers form E’s complement, 𝐸 = { 1, 3, 5, 7, 9 }; N = { E, 𝐸 } E 𝐸 1 Venn Diagram in Boolean algebra Represent the universe B = {0, 1} by a square. • {1} using shaded area (a) Constant 1 (b) Constant 0 x x 𝑥 (c) Variable x 𝑥 (d) 𝑥 Represent a Boolean variable x by a circle. • Area inside the circle -> x = 1; • Area outside the circle -> x = 0; 2 Venn Diagram – for two or more Boolean variables Represent x, y by drawing two overlapping circles x y (e) x×y x y (f) x+y x x y y z (g) x × y (h) x×y+z • 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 App: Verifying the equivalence of two expressions x y x y x y z z z (a)x (b) y+z (c)x× (y+z) x x x y y y z z z (d)x×y (e) x× z (f) x×y+x ×z Verification of distributive property x ∙ (y + z) = x ∙ y + x ∙ z 4 Another verification example x y x y x y x y z z z z x× y 𝑥 ×z y ×z 𝑥×𝑦+𝑥×𝑧+𝑦×𝑧 x y x y x y z z z x× y 𝑥 ×z 𝑥×𝑦+ 𝑥×𝑧 5 2.6 Synthesis using AND, OR, NOT gates • Can express the required behavior using a truth table Figure 2.15. A function to be synthesized. 6 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. f = x1x2 + 𝑥1 𝑥2 + 𝑥1 x2 7 x1 x2 f = x1x2 + 𝑥1 𝑥2 + 𝑥1 x2 = (x1x2 + 𝑥1 x2)+(𝑥1 𝑥2 +𝑥1 x2) = (x1+ 𝑥1 ) x2 + 𝑥1 (𝑥2 +x2) = 1 ∙ x2 + 𝑥1 ∙ 1 = x2 + 𝑥1 f (a) Canonical sum-of-products x1 f x2 (b) Minimal-cost realization Figure 2.16. Two implementations of a function in Figure 2.15. 8 Summary • To implement a function, – Use a product term (AND gate) for each row of the truth table for which the function is equal to 1. • If xi = 1 in the given row, xi is entered in the term; • If xi = 0, 𝑥𝑖 is entered in the term. – The sum of these product terms realizes the desired function • Different networks can realize a given function – Use algebraic manipulation to derive simplified logic expression, thus lower-cost networks. 9 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 mi 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 Figure 2.17 Three-variable minterms and maxterms. 11 Canonical SOP expression f = 𝑥1 𝑥2 x3 + x1𝑥2 𝑥3 + x1𝑥2 𝑥3+ x1x2𝑥3 Figure 2.18. A three-variable function. Manipulate f as following f = (x1+ 𝑥1 ) 𝑥2 x3 + x1(x2+ 𝑥2 ) 𝑥3 = 1 𝑥2 x3 + x1 1 𝑥3 = 𝑥2 x3 + x1𝑥3 A more concise form to specify the given canonical SOP expression (logical sum) f = (𝑚1, 𝑚4, 𝑚5, 𝑚6) = 𝑚(1,4,5,6) 12 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 Figure 2.17 Three-variable minterms and maxterms. 14 An example • The complement of a function 𝑓 can be represented by a sum of minterms for which f = 0. • 𝑓 = 𝑚2 = 𝑥1 𝑥2 • Complement this expression using DeMorgan’s theorem 𝑓 = 𝑓 = 𝑚2 = M2 = 𝑥1 𝑥2 = 𝑥1 + 𝑥2 15 𝑓 = m0 + m2 + m3 + m7 ∴ f = 𝑚0 + 𝑚2 + 𝑚3 + 𝑚7 = 𝑚0 ∙ 𝑚2 ∙ 𝑚3 ∙ 𝑚7 = M0 ∙ M2 ∙ M3 ∙ M7 = (x1+x2+x3) (x1+𝑥2 +x3) (x1+𝑥2 +𝑥3 ) (𝑥1 +𝑥2 +𝑥3 ) f = (x1+x3) (𝑥2 +𝑥3 ) Figure 2.18. A three-variable function. A more concise form to specify the given canonical POS expression (logical product) f= (M0, M2, M3, M7) = 𝑀(0,2,3,7) 16 • Cost of a logic circuit is – the total number of gates plus – the total number of inputs to all gates in the circuit. x2 f = 𝑥2 x3 + x1𝑥3 f x3 Cost = 13 x1 (a) A minimal sum-of-products realization Figure 2.19. Two realizations of a function in Figure 2.18. 17 x2 f x3 x1 (a) A minimal sum-of-products realization Cost = 13 x1 x3 f x2 (b) A minimal product-of-sums realization Figure 2.19. Two realizations of a function in Figure 2.18. 18 Example 2.3 Consider the function f(x1,x2,x3) = 𝑚(2, 3, 4, 6, 7) 1. Canonical SOP expression for the function f = m2+m3+m4+m6+m7 = 𝑥1 𝑥2 𝑥3 + 𝑥1 𝑥2 𝑥3 + 𝑥1 𝑥2 𝑥3 + 𝑥1 𝑥2𝑥3 + 𝑥1𝑥2𝑥3 2. Simplify the expression f = 𝑥1 𝑥2 + 𝑥1 𝑥3 + 𝑥1𝑥2 = 𝑥2 + 𝑥1 𝑥3 19 Example 2.4 Consider the function in Example 2.3, Specify it as a product of maxterms for which f = 0 f(x1,x2,x3) = 𝑀(0,1,5) 1. Canonical POS expression for the function f = M0M1M5 = (x1+x2+x3)(x1+x2+𝑥3 )(𝑥1 +x2+𝑥3 ) 2. Simplify the expression f = (x1+x2)(x2+𝑥3 ) = x2+x1𝑥3 20 Discussion (1) Given a logic function f(x1,x2,x3), 1. What is the index of the maxterm 𝑥1 𝑥2 𝑥3 ? Complemented entry -> 0 (𝑥1 𝑥2 𝑥3 ) -> uncomplement entry -> 1 0 1 0 (010)2 = 2 (decimal number) Therefore, 𝑥1 𝑥2 𝑥3 = m2 2. What is the logic expression of m5? (5)10 = (1 0 1)2 => m5 = (𝑥1𝑥2 𝑥3) 21 Discussion (2) Given a logic function f(x1,x2,x3), 1. What is the index of the maxterm (𝑥1 + 𝑥2 + Complemented entry -> 1 𝑥3 ) ? uncomplement entry -> 0 (𝑥1 + 𝑥2 + 𝑥3 ) -> 1 0 1 (101) = 5 (decimal) Therefore, (𝑥1 + 𝑥2 + 𝑥3 ) = M5 2. What is the logic expression of M5? (5)10 = (1 0 1)2 => M5 = (𝑥1 + 𝑥2 + 𝑥3 ) 22 Venn Diagram for Boolean algebra • Basic requirement for legal Venn diagram – Must be able to represent all minterms of a Boolean function Three variables Two variables x1 m2 m3 x1 x2 m1 m4 m5 m6 m7 x2 m2 m3 m1 m0 m0 x3 23 Venn Diagram for Boolean algebra • Basic requirement for legal Venn diagram – Must be able to represent all minterms of a Boolean function m5? m7? Three variables Two variables x1 m2 m3 m0 x2 m1 x2 x1 m4 m6 x3 m3 m1 m2 m0 24