# Bits necessary to allow for at least ten different choices

Digital theory
a. How many lines will be in a complete truth table for a three-input logic device? Show your reasoning or calculations.

b. How many bits are necessary to allow for at least ten different choices?

c. How many bits do two hexadecimal digits (for instance, 0xC3) represent? Why?

d. Create a NOR gate from AND and NOT gates (as many of each as you need.)

DeMorgan’s Theorem

Simplify the following expressions, using DeMorgan’s Theorem. (Simplified expressions should only have negation on single variables, not negated expressions.
I.E. (A + B) is okay but (A + B) would need to be simplified. )

Watch out for trick questions; just because a variable is in an expression, doesn’t mean it will be a part of the simplified answer! After all, (A + A) == 1 !
(Show your work or reasoning for partial credit.)

a. (A + B) * (A + B)

b. (A + B) + (C + D)

c. (A + B) + (A * B)
Karnaugh map minimization
Use a 3-variable Karnaugh map to minimize the following function. The final Boolean expression should use only ANDs and ORs (for instance, (E + F) etc). Assume that all variables are available as both positive and negative, if needed. You can use a truth table to construct the K-Map, if you want, but this is not required. Show your work (I.E. show how each term in the final expression maps to one or more 1s in the K-Map.)

Q = ABC + ABC + ABC + ABC + ABC + ABC
Practical considerations in Digital Design

a. Draw a circuit using one pull-down resistor (1k) and one SPST switch to provide a logical input to an inverter gate (shown). Show all voltage sources and sinks that you use.
b. Draw the truth table for the following function.
A
B
c. The above function can be simplified. Show the simplified equivalent schematic.

d. What pins (by pin number) are outputs on a 74LS32 Quad 2-input OR chip?
(See the chip pinout for the 74xx32 inside the cover of your book.)
Digital Arithmetic
a. Convert this hexadecimal expression to binary.
(Just convert the numbers; do not perform the subtraction yet): 5Ahex – 36hex.

b. Take the twos’ complement of the second binary number (the 36hex) from #b above.
c. Add this number to the first binary number from #a (the 5Ahex) to do the subtraction.
d. Convert the resulting number from binary to decimal.

e. Convert the resulting number from binary to hexadecimal.
Circuit design exercise
Create a circuit to implement a “parity tree” function.
This circuit takes three digital inputs (A, B, and C) and produces a single output, Q.

Q should be 1 when at an odd number of inputs are high, and 0 if an even number are high.

You may use as many AND, OR, and NOT gates as you like, but you may only use these types of gates for this exercise.

a.) Write a three-variable truth table for this function.

b.) Make a 3-variable Karnaugh map of your function.

c.) Circle the simplified terms on the Karnaugh map you created.

d.) From your circled terms, identify the simplified Boolean equation for the function.
e.) Draw a schematic of the circuit. Make sure to label A, B, C, and Q. Make sure AND and OR gates are clearly distinguishable (AND gates have a straight back; OR gates are concave at the back and have a sharper front.)

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