dmac05 wrote

I am only on the second page in Chapter 1 and already not understanding why the truth table has the values it has in the last column and how the boolean expressions are being created from it.

Which edition of the text are you using?

If you are talking about Figure 1.1 in the 1st Ed., then that is just an example of a Boolean function of three variables and the last column is the output of the function for each possible input. It is just an example and the values in the output are arbitrary -- the author could have (and perhaps did, though probably not) flipped a coin eight times to get the eight values.

For our purposes, we don't care what the function means. Perhaps (just making this up as I type), x represents the output of a thermostat, y represents the state of a switch, and x represents whether or not water is present in some tank. The person designing the system needs to use those three input to determine whether a heater should be turned on and, after doing a lot of analysis and tests, has come up with the last column of that table.

However that function came about, and whatever it means, we don't care. It's been thrown over the wall to us and we just have to implement it.

The text doesn't show, yet, how to go from the truth table (which defines how the function is supposed to behave) to the Boolean expression (which, hopefully, has the same behavior). Instead, it is essentially saying, "Hey, one of the kids next door thinks that this expression will work. Let's verify that it really does."

So, with that in mind, do you follow the steps used to verify that the equation really does produce the results in the table? If not, we can take a step back and work through that. If so, then if you would like to, we can look at one way (there are several) to come up with that equation starting from the table.