Completing Truth Tables
We're in the final section of 15-2. We will be completing Truth Tables. This is actually a continuation of the previous section where we were writing Boolean expressions. We started out with a circuit, we wrote a Boolean expression for it and now we'll be completing a Truth Table.
It may be helpful to build a Truth Table for an unfamiliar combinational logic circuit. It can also be a valuable troubleshooting tool. The Boolean expression must be stated in the correct form. If an expression is reduced to the point where there are no two variables joined by a vinculum - that means the NOT function - then there are only two types of expressions. They are either the Minterm or the Maxterm forms.
Minterm and Maxterm
The Minterm is often referred to as “the sum of the products.” The reason they say the sum of the products is because AC - this would be products and then the ORD function - here is the adding of them, so this is called the sum of products.
The Maxterm is the product of the sum. The product we'll be multiplying together but then we have the ORD function then it could look like this or like this. When you construct Truth Tables it often requires the Minterm type expression. If you have a Maxterm expression and you want to convert it to the Minterm expression, you would use Rule 36.
If you look back in your Rules on Boolean you can see that one ... I'll just perform that function. If we had A or B ended with C or D it would just be a matter of multiplying. This is something that you may have done in Algebra where you take A times C, A times D and then B times C, B times D. They have some rules about that. AC + AD and then it would it be + BC + BD. This is equivalent to that. With this expression, you can complete Truth Tables.
Completing Truth Tables
We had a couple of Truth Tables here. Here we have three variables Truth Table and with three there if you take two and you have three variables, then there are eight possibilities. We start out with 000 and we count up to 111 which would be seven and so that makes our eight variables.
We would get a high when these conditions are met, so A would be one, B would be one and C would be zero. It might be helpful just to turn that around. If we had 011 that would be in the form that we have it here, so 011 that's the number three. We should have a high right here and then this is going to be 000 and that was pretty straight, for we should have a high right here.
All of these other potential inputs will result in a zero and we will have a high at these two points. Then here's another one. This one we have four possible variables. In this case, we're going to have two to the fourth, so that would equate to 16 possibilities. That would go from zero to 15 and these are all the possible inputs that we could have for a four-variable digital circuit.
The first one is really easy. It's going to be 0000, so you can get a one if you had all zeros input. Then we will get a high also if we have ... this is going to be 0101 and if we flip it over to conform with that, we would have 1010 that is actually the number ten, 1010 so you should have a high right here.
Then AC, 1, 0, so actually with A and a C, that would be 0101, is that right? Let's see, 01, that would be the highest 01 right there. There the C is a one and A is one, that's it.
Then the rest of these ... this would be a zero, this would be a zero and all of these would be a zero. We should have an output at these three points and these three points will conform with those three.
The completing Truth Tables is fairly straightforward. Probably the hardest part ... you've got to draw the diagram here and just draw the correct number of possibilities. In this case, we had 16 and this one was 8. We only had 2, there would be 4. Then we looked at the Minterm and Maxterm and that completes our look at Truth Tables…