Please help!
Design a circuit using NAND gates that detects the “above” condition for two 2-bit values. That is, given two 2-bit variables x and y, F(x,y) = 1 when the unsigned integer value of x is less than the unsigned integer value of y. a) Give a truth table for the output of the circuit, F. b) Find a minimal sum of products for F. c) use either a PLA or PAL.

Question

Please help! 
 Design a circuit using NAND gates that detects the “above” condition for two 2-bit values. That is, given two 2-bit variables x and y, F(x,y) = 1 when the unsigned integer value of x is less than the unsigned integer value of y. a) Give a truth table for the output of the circuit, F. b) Find a minimal sum of products for F. c) use either a PLA or PAL.

anonymous · Answer

okay, so let's write a truth table for our F(x,y) = 1

$$\begin{array}{c|c}
x&y&F\
\hline
00 & 00 & 0 \
00 & 01 & 1\
00 & 10 & 1 \
00 & 11 & 1\
01 & 00 & 0\ 
01 & 01 & 0\
01 & 10 & 1 \
01 & 11 & 1\
10 & 00 & 0\
10 & 01 & 0\
10 & 10 & 0\
10 & 11 & 1\
11 & 00 & 0\
11 & 01 & 0\
11 & 10 & 0\
11 & 11 & 0
\end{array}$$

anonymous · Answer

we can write a sum-of-products from the truth table easily (hint: look at all the columns that yield $F(x,y)=1$), if we split $x,y$ into individual inputs for each bits -- $x_1, x_2, y_1, y_2$ so: $$(x_1'x_2'y_1'y_2)+(x_1'x_2'y_1y_2')+(x_1'x_2'y_1y_2)+(x_1'x_2y_1y_2')+(x_1'x_2y_1y_2)+\dots$$

and then minimizing the expression is easy once you have ita ll out