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

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

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

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

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

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

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

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

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

Summary 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 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

Figure 2.17 Three-variable minterms and maxterms. 11

Figure 2.18. A three-variable function. 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 15

Figure 2.18. A three-variable function. 16

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

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

Example 2.3 19

Example 2.4 20

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

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

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

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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