00:01
So to solve these functions, it's with these variables, it's analogous to using truth tables for logic.
00:10
So how we should set these up is that for part a, for example, we know that we're going to have a variable x, a variable y.
00:20
Then we're going to know we're going to need to take the complement of x, so we're going to have x complement.
00:26
And then the final step is we're going to know we're going to need to evaluate this whole impression.
00:30
Expression together so x complement product y so this is a product in between them so with that in mind we can go ahead and solve this so what we know is that we are going to need to use it's a function of three variables but we're only really using two in here so we can add this third variable but we know that we're only going to need two so so what happens is that we know that x can have either be 0 or 1.
01:11
And with a function of three variables, we're going to have eight total combinations, which is just 2 to the third power, because 3 represents our number of variables.
01:23
So we're going to have 4 zeros and 4 1s.
01:33
All right.
01:34
And the same thing for why.
01:36
So it's going to need 4 zeros in total.
01:42
And 4 -1s.
01:46
0 -0 -1.
01:49
And the last thing for z is the same thing.
01:52
And with this part, we're just going to alternate between 0s and 1.
02:02
All right.
02:03
Now with that in place, we can take x complement.
02:07
So it's just going to switch whatever state the x is in.
02:10
So we're going to have 4 -1s up top, and then 4 -0s.
02:19
Then let's evaluate the whole thing, which is x complement, product, so x complement product y is both of these need to be one for it to be a one otherwise it's a zero so it's going to be zero zero and then third one it's one and one so it's going to be one and then their fourth one is one and one also it's going to be one and then we just have all zeros on the x complement so these are all going to be zeros as well so that's pretty much it now we can move on to part b and so set it up the same where we know we're going to have a function of three variables.
03:04
Let's write down our variables.
03:06
Our variables are x, y, and c.
03:11
Then at some point we know we're going to need to perform the operation y product z.
03:17
And then after that, we can evaluate the whole thing.
03:20
So x, sum, y, product, z.
03:26
So same schema.
03:28
So we're going to start by recognizing that we're going to need two to the power.
03:32
Our three rows.
03:35
So best practice is to start with one, two, three, four.
03:41
So half of them are zero here, half of them are one.
03:46
Then for the second variable, what you do is just half of them are still zero, half of them are still one, but you switch every fourth.
03:55
So zero one, zero one, zero one.
04:02
And then for the third variable, you switch every fourth.
04:04
Switch every eighth so that's going to be every term so zero one zero one zero one zero one all right with that in place we can start evaluating things so why product is eight both of these terms need to be one or to be a one so what happens here is that the first three going to be zero then we have a one and a one so that's going to be a one and zero 0 0 to the last term which is just 1 that we can evaluate the whole thing together so wherever this is 0 0 for the first one so 0 0 0 for the second one 0 same for the third one but on the fourth one and it's the y product z is 1 so it's 1 and then for everything below that the x is 1 so it's all going to be 1 so we have one, two, three, four ones.
05:11
And that's the finished expression for that.
05:16
And you can move on to part c.
05:18
So same concept where we're going to have three variables, but the calculations here are a little bit more intense.
05:25
So we're going to have three variables, but we know at some point we're going to need to calculate the complement of y.
05:33
You know at some point we're going to calculate the product of x complement y.
05:40
Then we're also going to need to calculate the x product y if we want to just make sure we only have two terms then x product y product z then x product y product z the complement of that whole thing and then at the end our full expression so x complement y plus x y z then the complement of that whole thing.
06:23
So that's it.
06:24
Just to make it easier to follow, i'll add vertical columns.
06:43
Okay, start with x.
06:45
We know we're going to need four zeros, four one, following the same pattern as we've used before.
06:57
So i'll just fill that out.
07:06
Same thing for z, same pattern we've been using.
07:18
And then y complement is flip whatever y was.
07:21
So 1 .1 .0.
07:28
0110 0.
07:30
Then x product y complement we need to look at both the x and y complement so here for the first four we can just kind of create a short circuit we know that all of these are all the x's are zero so you know the first four at least are going to all be zero then you need to look at them so all the x's here are going to be one so as long as the y complement is also one which is only for these two, then it's going to be one.
08:05
Now let's look at the x product y.
08:08
So like save short circuit evaluation, we know the first four xs are zero, so the first four x product y is going to be zero...