Boolean Algebra for 1.9 Demulitplexor

[rich]

I'm trying to do the boolean logic so I can create my K-graph (not in the book, but helpful!) to figure out the circuits. I don't understand how to work from a truth table based on the data table in figure 1.9.

What am I solving for when there's 2 outputs?

I'm not sure where to begin with the algebra. I appreciate any insights.

[cadet1620]

The expanded truth table for mux will have inputs sel and in and outputs a and b:

sel|in || a | b
---|---||---|---
 0 | 0 || 0 | 0
 0 | 1 || 1 | 0
 1 | 0 || 0 | 0
 1 | 1 || 0 | 1 

Since there are two outputs, you need two Karnaugh maps, one for each output. Since there are only two variables, they are going to be 2 by 2 k-maps, which are not too interesting.

When you have more than 2 inputs, then things become more interesting. For the optimum solution you need to start by looking for minterms that are common to both k-maps.

Say we have a part with three inputs a, b, and c, and two outputs F and G whose k-maps are:

   |       bc              |       bc
 F | 00  01  11  10      G | 00  01  11  10
---+---+---+---+---+    ---+---+---+---+---+
 0 | 0 | 1 | 0 | 0 |     0 | 1 | 1 | 0 | 1 |
a  +---+---+---+---+    a  +---+---+---+---+
 1 | 0 | 1 | 1 | 1 |     1 | 0 | 1 | 0 | 0 |
   +---+---+---+---+       +---+---+---+---+

F and G have the minterm (~b)c in common so that Not and And gate can be shared with the circuits for both outputs.

      F = ab + (~b)c
      G = (~a)(~c) + (~b)c


This can get complex rather quickly.  Especially when you add "don't care" values to the functions.

--Mark