# Boolean Expressions and Truth Tables

#### Digital

We're in Boolean Expressions and Truth Tables. Being able to write the Boolean expression and completing the truth table is an important skill in troubleshooting digital systems. Writing Boolean expressions begins with starting at the inputs and working toward the outputs. The expression used to describe the output of each gate is determined using the basic rules of logic gate expression.

In this section, we will be writing the Boolean expression. In the next section, we'll actually be completing the truth tables. What we have here are several circuits and these are all circuits ... by the way may come from your multi-sim package. You can pull your circuits up and activate them in multi-sim. When you do so, if you'll simply press the letter A, B or C, you will select the logic levels that are going to these gates, but our purpose here is to determine the Boolean expression. This particular one is from circuit 15_17.

First, we need to evaluate the inputs. We have input A, we have input B and we have input C and so A and B coming to this AND gates, so out of the AND gate we'll have AB. That will come into this OR gate here. In fact, the process as we mentioned earlier is you started the inputs and you move to the output. Here we have evaluated ... this is an input, this is an input and they're combined at the output in this OR gate, so what we would see in the OR gate as the final output would be AB + C.

What that means is that if we had a valid high A and B we would get an A output and if we had a valid input from C, we would also get an output as what this expression actually means. This represents our Boolean expression for this circuit.

Here we have another one. Inputs are A, B, C and D. Out of the AND gate here we're going to have AB and then the OR gate here. We're going to have C + D and then these two inputs are going to be AN or NAND together at the output, so we're going to have AB and we're going to have C or D and it is going to be inverted like so.

As well as evaluating the expression, we also want to simplify them if it is possible. From previous lessons, we seemed to simplify this. We need to break the NOT symbols that cover more than one variable, so we're going to break it here and we will break it here and we will break it here.

What that will give us - we'll have A NOT and we'll have to change the AND to an OR, B NOT and then we will have C NOT, D NOT. This should become a plus here because we're breaking ... sorry about that. We have A NOT + B NOT. Again, we have A NOT + B NOT + C NOT, D NOT. Again, if we had any of those we should get enough. Any of this combination, either A NOT or B NOT or C NOT, D NOT, should illuminate our light bulb.

Next one, we have A. Let's make sure we keep these drawings. We have a B and it is coming in here. I mean A goes to two different places, so we need to make sure we keep that straight A and then this would be the C input here. Let's see, we come into an inverter so that would be a NOT. Then coming into this NAND gate we have A NOT, B but since it's notted, that's going to invert the entire output and then here we go. We have a C NOT and then in this NAND gate we're going to have an output of A, C NOT and then that entire output is inverted. Then we're going to NAND both of these together, so we would have A NOT, B NOT ended with A, C NOT. Then we have the final inverter here, so the entire equation would be inverted.

Again, we're going to need to simplify this. Let's place it here, here and here. This is going to simplify to A NOT, B and then we'll put a four symbol here because it's only a single not and this would be A, C NOT. That should represent our final expression of the circuit.

Here, we have another. Again, inputs A, B, C and D and A + B here, A or B and we're going to NOT it and then down here we will have C NOT and so out of this NOR gate we'll have C NOT or D and the entire expression will be notted. Then we'll NOR them together, so we would have A + B NOT or C NOT or D and then we have an inverter here, so the entire equation would be NOT and then we're going to simplify or split it here, here and here.

Let's see, this has got a double NOT on it, so this would just remain the same, A + B. We have a single NOT here, so this is going to change function and this one, let's see ... this has got double NOTs so we're going to retain the C NOT + D and that should result in the final C. We have a NOT here and a double NOT here, so that is ... Okay, we have one NOT here, actually, there are three NOTs, so we get that. That should do it, so this represents our final expression of that one.

Here we have inputs A, B and C and actually, this is an exclusive NOR. This is an exclusive OR but it's inverted and so we will have A. This is the symbol for exclusive OR, A or B but it's inverted. We're going to put it in parenthesis. Then we have C and this would become C NOT and then they're ended together here. We're going to have A exclusive or B but this is going to be exclusive NOR and that's going to be AND with C NOT. Then the entire equation, since we're AND because we got a C NOT here, we got A and B NOT and then we're ending it and the entire equation is going to be notted.

As a result, this will remain the same. We have A, exclusive OR, B and then C since there ... Let's see this, it's going to be changed since there's only one here. We're going to break it here. No, we won't break that one. We'll need to break it here, so this would become a plus. Here we would have an ultimate result here, we had A or B or C. We would not break this because we've got a double NOT here.

Final one, here we have A, B and C and here we're going to have an A or B and that's going to be exclusive OR with C, so we're going to have A or B or C and so this will just simplify to A or B or C. We have no NOTs, so that should be the final expression for that one.

These are a few Boolean expressions. We took some circuits, we wrote Boolean expressions and then we simplified them. We did this for a number of circuits. Go through this and there are more practice problems and there is the homework for your enjoyment.