(1) What does the boolean expression
    (NOT ((((NOT X) AND Y) OR (X AND Y)) AND (((NOT X) AND (NOT Y)) OR (X AND (NOT Y)))) i.e. ((x'.y + x.y).(x'.y' + x.y'))' simplify into?
    Give the truth table for this function.

(2)  Draw a logic circuit with AND, OR and NOT gates to implement the
     following truth table. In addition to using the inputs i, s0 and s1, 
     you may also feed hardwired 1's and 0's to any of the gates' inputs,
     if that is what is required to implement this truth table.
     
      i s0 s1 || o0 o1 o2 o3
     -----------------------
      0 0  0  || 0  0  0  0
      0 0  1  || 0  0  0  0
      0 1  0  || 0  0  0  0
      0 1  1  || 0  0  0  0
      1 0  0  || 1  0  0  0
      1 0  1  || 0  0  1  0
      1 1  0  || 0  1  0  0
      1 1  1  || 0  0  0  1


      Give a Boolean expression for each of the outputs o0, o1, o2 and o3
      in terms of the inputs i, s0 and s1.

      Extra Credt: Can you describe in a few words what this circuit does?
      (Hint: How do the outputs o0, o1 and o2 relate to the inputs i, s0 
       and s1?)



(3) A gate is called "universal" if you can implement any boolean function
    with only gates of that kind. In this problem, you will show that the
    NAND gate is universal.

    Recall that since we already know that together, AND, OR, and NOT
    gates are universal, all we need to do is show how to make these
    three gates out of NAND gates.

    The NAND gate has the following truth table:

    X Y || NAND(X,Y)
--------------------
    0 0 || 1
    0 1 || 1
    1 0 || 1
    1 1 || 0

    You can draw a NAND gate as just a box with "NAND" written inside.

    (a) Make a NOT gate out of a NAND gate.
        Recall that you can "hard wire" an input to a NAND
        gate to always be 0 or always be 1.

    (b) Make an AND gate out of NAND gates.  You might want to
        use what you did in part (a) here.

    (c) Make an OR gate out of NAND gates.  Again, part (a) will
        be helpful.

    You've just shown how to make a circuit for *any* truth table
    out of just NAND gates!


(4)
     A "transistor" is just a gate with the following truth table:

     X Y || T(X,Y)
------------------
     0 0 || 0  
     0 1 || 1
     1 0 || 0
     1 1 || 0

     Note that the transistor is a little different from the gates
     we've seen so far, because it is "asymmetric."  It treats
     its two inputs differently, unlike AND, OR, and NAND gates.

     In what follows, it is worthwhile to have some intuition into
     transistors.  The output of the transistor is just equal to Y,
     unless it is "blocked" by X:  that is, if X is 1, then the output
     is always zero.  Otherwise, the output is just Y.

     Show that transistors can be used to implement any boolean
     function, ie, a transistor is "universal".


        Hints:
        i)  You may need more than 1 transistor.
        ii) You are allowed to use inputs that are always fixed at either 0 or 1