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Logic Gates and Equivalence Proofs - Prof. Fenghui Yao, Assignments of Information Technology

Tennessee State University (TSU)Information Technology

Prof. Fenghui Yao

Problems related to logic gates and their equivalence. It includes steps to build and, or, and not gates using nor and 2-to-1 multiplexer gates. Additionally, it provides a proof of equivalence between pairs of logic expressions using both truth tables and symbolic manipulation.

Typology: Assignments

Pre 2010

Uploaded on 08/16/2009

koofers-user-ok3-1 🇺🇸

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P17 Problem 1.1

d. Show that the NOR gate is universal.

e. Show that the 2-to-1 multiplexer is universal.

f. Is there any other two-input element, besides NAND and NOR, that is universal?

Answer to d. Show how to build AND, OR, and NOT gate from NOR gate.

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Download Logic Gates and Equivalence Proofs - Prof. Fenghui Yao and more Assignments Information Technology in PDF only on Docsity!

P17 Problem 1. d. Show that the NOR gate is universal. e. Show that the 2-to-1 multiplexer is universal. f. Is there any other two-input element, besides NAND and NOR, that is universal? Answer to d. Show how to build AND, OR, and NOT gate from NOR gate.

Answer to e. Show how to build AND, OR, and NOT gate from 2-to-1 multiplexer. F  B A  B  BA  B  A  A ( 1  B )  B  A  B Answer to f. Yes,

Circuit diagram: e0 x1x 00 01 11 10 x3x2 (^00 1 0 1 ) 01 0 1 1 1 11 x x x x 10 1 1 x x e 0 (^)  x 3  x 1  x 2 x 0  x 2 x 0 x3x2x1x e e e e e e e

P18 Problem 1. Prove that the following pairs of logic expressions are equivalent, first by truth table, and then by means of symbolic manipulation. Proof: RHS  xyz  xyz  xyz  xy z  xyz  xyz  xyz  xy z ( xy  xy ) z ( xy  xy ) z ( x  y ) z ( x  y ) z  x  y  z  LHS Truth table: Input xyz LHS x  y  z RHS xyz  xyz  xyz  xy z 000 0 0 001 1 1 010 1 1 011 0 0 100 1 1 101 0 0 110 0 0 111 1 1

Truth table: Input wxyz LHS RHS 0000 0 0 0001 0 0 0010 1 1 0011 0 0 0100 0 0 0101 1 1 0110 1 1 0111 1 1 1000 1 1 1001 0 0 1010 1 1 1011 0 0 1100 1 1 1101 1 1 1110 0 0 1111 1 1

p.18 Problem 1.

Assume x = (x 2 x 1 x 0 ) two and (y 2 y 1 y 0 ) two are 3-bit unsigned binary numbers. Write down a logic

expression in terms of the six Booleans variables x 2 , x 1 , x 0 , y 2 , y 1 , and y 0 that assumes the value 1

iff:

a. x = y

b. x < y

c. x ≤ y

Answer to a :

f 1 (^)  x 2  y 2  x 1  y 1  x 0  y 0

Answer to c :

2 2 1 1 0 0 2 2 2 2 11 2 2 1 1 0 0 3 1 2 x y x y x y yx x y yx x y x y y x f f f                 