Fordham, New York City's Jesuit University
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Fordham College Lincoln Center Computer Organization
Dept. of Computer and Info. Science Spring, 2005

Homework Assignment 1
Due date: January 31

In addition to the basic laws we discussed on pages B-4 and B-5, there are two important theorems, called DeMorgan's theorems:

\begin{displaymath}\overline{A + B} = \overline{A} \cdot \overline{B} \quad\mbox{and}\quad
\overline{A \cdot B} = \overline{A} + \overline{B} \end{displaymath}

[10] $<$§B.2$>$ Prove DeMorgan's Laws by constructing a truth table. Show intermediate steps.

[15] $<$§B.2$>$ Prove that the following two equations for the function $E$ are equivalent, by using DeMorgan's laws and the axioms shown on page B-6.

\begin{displaymath}E = (A \cdot B + A \cdot C + B \cdot C) \cdot (\overline{A
\cdot B \cdot C})\end{displaymath}

\begin{displaymath}E = A \cdot B \cdot \overline{C} + A \cdot \overline{B} \cdot C
+ \overline{A} \cdot B \cdot C\end{displaymath}

[10] $<$§B.2$>$ One logic function that is used for a variety of purposes (including within adders and to compute parity) is exclusive OR. The output of a two-input exclusive OR function is true only if exactly one of the inputs is true. Show the truth table for a two-input exclusive OR function and implement this function using AND gates, OR gates, and inverters.

[10] $<$§B.2$>$ Show that the exclusive OR is associative. That is,

\begin{displaymath}(A \oplus B) \oplus C = A \oplus (B \oplus

where $\oplus$ denotes the excusive-OR operation. Hint: the proof can be done by constructing a truth table for each side of the equation.

[15] $<$§B.2$>$ Prove that the NOR gate is universal by showing how to build the AND, OR, and NOT functions using only two-input NOR gates.

[15] $<$§§B.2,B.3$>$ Using a diagram in the same style as the one in Figure B.3.5, show how to implement a full-adder using a PLA.

Robert Moniot 2005-01-24