EXERCISES for Week 5



 1. Draw a Boolean circuit using AND, OR, and NOT gates that implements
    the function X(Y + Z).


 2. Draw a Boolean circuit using AND, OR, and NOT gates that implements
    the function XY + XZ.


 3. Draw a Boolean circuit using AND, OR, and NOT gates that implements
    the function of three variables that is 0 if precisely two of the
    variables have the same value, 1 otherwise.


 4. Draw a Boolean circuit using AND, OR, and NOT gates that implements
    the function of four variables that is 0 if precisely two of the
    variables have the same value, 1 otherwise.


 5. A NOR gate is a logic element with two inputs that produces 1 if the
    inputs are both 0, 0 otherwise. Logic circuits can be built 
    using NOR gates (no other kind of gate).  Show how to implement
    a NOR gate with two transistors.


 6. Draw a logic circuit that uses NOR gates to implement an
    AND circuit (output 1 if both inputs are 1, 0 otherwise). 


 7. What is the effect of wiring the output of a D flip-flop to its
    input?


 8. What is the result of computing the AND of all the output lines of a
    decoder?  Answer the same question for OR.


 9. What is the result of wiring the output of a decoder to the input of
    a multiplexer, using the same address lines (see diagram below)?























                    Answers to Exercises for Week 5

 1. 



 2. 



 3. Unless all three are 0 or all three are 1, exactly
    two must be the same.  It's simpler to compute (NOT f), as follows:
 
       x y z  f ~f
       0 0 0  0  1      ~f = ~x~y~z + xyz
       0 0 1  1  0
       0 1 0  1  0
       0 1 1  1  0
       1 0 0  1  0
       1 0 1  1  0
       1 1 0  1  0
       1 1 1  0  1
 
 4. 
       x y z w  f       x y z w  f
       0 0 0 0  0       1 0 0 0  0
       0 0 0 1  0       1 0 0 1  1
       0 0 1 0  0       1 0 1 0  1
       0 0 1 1  1       1 0 1 1  0
       0 1 0 0  0       1 1 0 0  1
       0 1 0 1  1       1 1 0 1  0
       0 1 1 0  1       1 1 1 0  0
       0 1 1 1  0       1 1 1 1  0

  f = ~x~yzw + ~xy~zw + ~xyz~w + x~y~zw + x~yz~w + xy~z~w

 5. 





 6. 





 7. It gets stuck at whatever the initial value of the input was.  If 0,
    then the first clock pulse produces "reset" which keeps the input at 0.
    If 1, then the first clock pulse produces "set" which keeps the input at 1.

 8. Exactly one of the outputs is 1, and the rest 0, so the AND is always
    0 and the OR is always 1.

 9. Always produces 1.  The decoder produces 1 on whatever output line is
    addressed; the multiplexer selects that line.