- (10 points) Draw the truth table for the circuit pictured here:
- (10 points) A function F on 3 variables
is given by:
F(a,b,c) = a'b'c + a'bc + abc + abc'
This function can be minimized to two terms. Determine a minimal function
Fmin that produces the same output (i.e. is equivalent to) the function F
(you should be able to determine a minimal function by inspection).
- (10 points)
Start with the same function F again, as defined in Problem 2. Show how to
obtain the minimized
function Fmin by applying the
postulates and theorems
of Boolean Algebra (the postulates (P1 through P6) and theorems (Th1 through
Th16) are listed on pages 2 and 3 of this .pdf file; the numbers on the bottom of the pages
are 44 and 45). Here are the first couple of steps to get you
started. Don't combine any steps (each step should follow from a single application
of one of the postulates or theorems).
F(a,b,c) = a'b'c + a'bc + abc + abc' given
F(a,b,c) = a'(b'c + bc) + abc + abc' P4 (distributive law)
. . .
- (10 points) In class we showed that AND and NOT are functionally complete. We then
showed that NAND is also functionally complete, by designing the functions AND
and NOT using only NAND gates.
Draw a logic diagram for a circuit that computes A OR B. Your circuit may
use only NAND gates.
- (10 points) In class we saw how to construct a black box representation of
a 4-to-1 MUX built from three 2-to-1 MUXes.
Suppose you want a MUX where you select 2 bits at a time, instead of 1 bit. For example, you might want to select either of the inputs x1x0 or
y1y0. Since you're only selecting one of two inputs, you should still use only
one control bit, but now instead of 1 output bit, you will have 2.
Draw a black box representation of a 2-to-1 two-bit MUX that is implemented using two 2-to-1 one-bit MUXes. Follow the conventions we used in class for
representing black boxes.
- (15 points) Do problem 3.24 from the textbook.
- (15 points)
- (20 points) The Muller C-Element is a sequential circuit that behaves as follows:
the output of the circuit reflects the inputs when the states of all inputs
match. The output remains in this state until the inputs all transition to
the other state. Here is a characteristic table for a two-input Muller
A B | Q(t+)
0 0 | 0
0 1 | Q(t)
1 0 | Q(t)
1 1 | 1
Draw a finite state machine for the 2-input Muller C-Element.
- b). Create the state transition table for the finite state machine in