SP10-pp1-sol

SP10-pp1-sol - UNIVERSITY OF CALIFORNIADAVIS DEPARTMENT OF...

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UC Davis 1 Hussain Al-Asaad U NIVERSITY OF C ALIFORNIA —D AVIS D EPARTMENT OF E LECTRICAL & C OMPUTER E NGINEERING EEC180A—D IGITAL S YSTEMS I Spring 20 10 S OLUTIONS OF P RACTICE P ROBLEMS — S ET 1 1. Draw the schematics of in terms of AND, OR, and inverter gates. 2. Draw the schematics of the following functions in terms of NAND and inverter gates. 3. (a) Prove the simplification theorem using the first eight laws of Boolean algebra (page 51 of textbook). There are several answers to this problem. Below is one of them. = Distributive law (8) Distributive law (8) Complementarity theorem (5D) Operations with 0 (1) Idempotent theorem (3D) Commutative law (6D) FW X Y Z + () = W X Y Z F X Y Z a) XYZ [] = X Y Z b) XY XZ + + == + + X = + + XX Y + YX Y + + XX YX YY + ++ 0 + X Y
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UC Davis 2 Hussain Al-Asaad Distributive law (8) Commutative law (6D) Complementarity theorem (5) Operations with 1 (1D) Idempotent theorem (3) (b) Prove the simplification theorem using the first eight laws of Boolean algebra (page 51 of textbook). There are several answers to this problem. Below is one of them. = Distributive law (8) Commutative law (6D) Distributive law (8) Commutative law (6D) Complementarity theorem (5D) Commutative law (6) Operations with 0 and 1 (1) Operations with 0 and 1 (1D) Complementarity theorem (5) Distributive law (8) Commutative and associative laws (6, 7) Commutative and associative laws (6D, 7D) Distributive law (8) Operations with 0 and 1 (2) Operations with 0 and 1 (1D) 4. (a) Use DeMorgan’s theorem to compute the complement of . (b) Use DeMorgan’s theorem to compute the complement of . 5. Using Boolean algebra, verify that the schematic below implements the XOR function. Using the NAND equation for every gate we get: , , , and . Substituting the values of variables A , B , C in terms of X and Y in the function F , and using Boolean algebra, we get: . 6. Simplify the function using the theorems of boolean algebra. Write the particular law XX YY + () + + + 1 + + X XY + XZ + + = + + + XXY + Z + XX Y + ZX Y + + ZX ZY +++ 0 0 ++ 1 X X + ZYX + Z Y Y Y + X Z YX Y Y + XZ Y 1 + XY Z 1 + + 1 1 + + FA B C D + = B C D + ABC D + + D + ABCD + + == FX Y Z + = Y Z + XYZ + + === = X Y F A B C AX Y = BA X = CA Y = FB C = C A X AY + AX Y + + + + + = = = = = = FXY , + + =
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UC Davis 3 Hussain Al-Asaad or theorem you are using in each step. For each simplified function, give the number of literals. = (4 literals) (Distributive law— 8) (6 literals) (Commutative law—6D) (6 literals) (Distributive law— 8) (8 literals) (Idempotent theorem— 3D) (7 literals) (Commutative law—6D) (7 literals) (Complementarity theorem—5D) (5 literals) (Operations with 0 and 1—1) (5 literals) (Commutative and associative laws—6, 7) (5 literals) (Commutative law—6D) (5 literals) (Distributive law— 8) (4 literals) (Operations with 0 and 1—2) (3 literals) (Operations with 0 and 1—1D) (3 literals) (Distributive law— 8) (2 literals) (Operations with 0 and 1—2) (1 literal) (Operations with 0 and 1—1D) (1 literal) 7. Given the following function in product of sums form (not necessarily minimized) (a) Express the function on the canonical sum of products form using “little m” notation. The simplest way to solve this part is to complete the truth table. For every row of the truth table, compute F . For example,
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This note was uploaded on 04/25/2011 for the course EEC 180A taught by Professor Redinbo during the Spring '08 term at UC Davis.

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SP10-pp1-sol - UNIVERSITY OF CALIFORNIADAVIS DEPARTMENT OF...

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