1. Draw the membership function for the following Fuzzy expressions
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1 1. Draw the membership function for the following Fuzzy expressions a. A OR B b. NOT (B AND C) c. A OR (B AND C) Answers to Practice Problems Practice problems - Deriving Boolean Expressions From Truth Tables: What is the Boolean expression for the following truth tables? 1. A B f b. A B C f Answers: a. A B +A B+AB b. A B C+A BC +AB C +ABC Practice problems - Deriving the Truth Table for a given function Show the truth tables for the following functions 1. f = AB 2. f = A B + C 3. f = B 4. f = 1 (2 variables) 5. f = 0 (3 variables) NTC 2/26/05 95
2 6. f = A + B + C 7. f = AB + A C 1. A B f 2. A B C f A B f 4. A B f A B C f6. A B C f A B C f NTC 2/26/05 96
3 Practice Problems - Facts 1. How many minterms exist for a Boolean space of 5 variables? Ans: How many possible Boolean functions of 4 variables are there? Ans:64K 3. List all minterms for a Boolean space of 1 variable. Ans: A, A 4. List all functions for a boolean space of 1 variable Ans: 0, A, A, 1 5. If f = A B, what is the DNF for f? Ans: f = AB + AB + A B Practice Problems - Algebraic Simplification 1. Write the duals of the following Boolean equations a. A + B = 1 AB = 0 b. A(B +C ) = 0 A+B C = 1 c. A B + B C = 1 (A +B)(B +C) = 0 d. (A + B )CD = 0 AB +C+D = 1 2. Simplify Algebraically the following Boolean expressions a. f 1 = A + A B = A + B b. f 2 = A(A+B ) = A c. f 3 = A + ABC +A C = A + C d. f 4 = AB + AB + A B + A B = 1 e. f 5 = A(B + C)A = 0 f. f 6 = (A + B)(A + B ) = AB + A B g. f 7 = (A B + C) + C = A + B + C Practice Problems - Canonical Forms 1. Find the Disjunctive Normal Form for the following Boolean expressions, assuming the DNF has the number of variables given. a. f a = A + B = AB + AB + A B b. f b = A + B = ABC + ABC + AB C + AB C + A B C + A B C c. f c = AB C Already DNF in three variables NTC 2/26/05 97
4 d. f d = 1 = ABC + ABC + AB C + AB C + A B C + A B C + A BC + A BC e. f e = AB + A C + BC = AB C+ AB C +A BC + ABC + A B C + A BC f. f f = (A + B)(A + C) = ABC + AB C + A BC + A BC 2. Find the Conjunctive Normal Form for the following Boolean expressions, assuming the CNF has the number of variables given. a. g a = A + B Already CNF in two variables b. g b = A + B (A + B + C)(A + B + C ) c. g c = ABC + ABC + AB C + A B C = (A + B + C )(A + B + C)(A + B + C)(A + B + C ) d. g d = (A + B )(B + C ) = (A + B + C)(A + B + C )(A + B + C )(A + B + C ) e. g e = 1 = (A + B)(A + B )(A + B)(A + B ) Practice problems-karnaugh Maps 1. Write the simplified function given by the following truth tables: a. A A Answer: b. f = A + B A A Answer: f = 1 NTC 2/26/05 98
5 2. Show the Karnaugh Maps and simplify the following Boolean expressions: a. A (B + C )+ BC A A Ans: A + BC b. AB + B (C + A D) + (B + CD) Sum of Products form = ab + B c + A B D + B + CD C D C D CD CD A B A B AB AB Ans: = B + C + A D c. AB + AB A B + A B B B A 1 1 A 1 1 Ans: 1 d. ABC + ABC + AB C + AB C + A BC + A BC + A B C NTC 2/26/05 99
6 A A Ans: A + B + C Practice problems - Exclusive-Or 1. Identify each of the following truth tables is for an XOR function, a NOT XOR function, or neither. If the function is an XOR or NOT XOR function, write the corresponding boolean expression using only the XOR operator (). a. ABC f b. AB f c. ABC f d. AB f ABC AB Not xor (AB) 2. Which of the following Boolean expressions represents an Exclusive-Or or a NOT XOR (can be written using only the XOR and NOT operators)? For those that do, rewrite the expression using only XOR and NOT operators a. AB +A B AB b. AB+A B (AB) c. A B+AB NO d. ABC+AB C +A BC +A B C ABC e. A BC+AB C +ABC +A B C NO NTC 2/26/05 100
7 Practice Problems - NANDs and NORs 1. Assume the following symbolism: A nand B = AB, A nor B =. Evaluate the following Boolean expressions: a. A(A1) Ans: 1 b. B0 Ans: 1 c. (A0)1 Ans: A d. B(01) Ans: 1 2. Rewrite the following Boolean expressions (written in the NAND/NOR symbolism above) into expressions using traditional AND, OR and NOT symbolism. a. (AB)C Ans: ((A+B) C) = (A+B) + C = A+B+C b. A((AC)0)Ans: (A((A+C) 0) ) = A c. (AB)C Ans: ((AB) C) = AB+C d. (B1)1 Ans: 0 3. Convert the following functions into expressions using only NANDs and also into expressions using only NORs a. AB+C c. (A+B)C b. ABC d. A+B+C Practice Problems - Logic Circuits 1. Draw the logic circuits corresponding to each of the following Boolean expressions. Do not simplify the expressions. a. AB+C(A +B) b. A+B+C(A +C ) c. (A+B )(BC +A)+D NTC 2/26/05 101
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