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Welcome to the last video. In[br]the Boolean series. In this

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video we will look at some of[br]the more complicated Boolean

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expressions and therethrough[br]tables.

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Now let's make matters a[br]little bit more complicated.

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By using more brackets.

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P&R

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or[br]Q&R.

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If then. P or not Q first[br]thing that we want to establish

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is to find out how many[br]different inputs are there.

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BRQRPQ there are[br]altogether three different

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letters, so P.

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Q.

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R.

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Again, three different letters.[br]Two to three gives me 8, so I'm

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going to have eight different[br]input combinations or eight

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different rows in the truth[br]table. Half of it is 4, so I'm

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going to start with four truths[br]and then four forces.

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Then half of four is 2, so I'm[br]going to have two 2, two fours

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until I fill up the table.

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Then just use true force.

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The force to fill[br]in the last column.

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Now let's look at the different[br]brackets. I have got a square

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bracket here. Which is combining[br]together this bracket using the

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event operation. So to be able[br]to use the if then operation

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first I need to know what is the[br]value inside the square bracket,

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but unfortunately inside the[br]square bracket I have got two

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different brackets. So first I[br]need to calculate the P&R.

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Then I need to calculate[br]the Q&R.

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Then I will be able to calculate[br]what are the values in the

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square bracket, not to save some[br]space. I'm just going to put the

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square brackets sign into here.

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Now once I've got the square[br]bracket, I need to combine the

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values of the square bracket[br]with this bracket, but inside

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this bracket I have got an not Q[br]as well. So first I will need to

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calculate the value of not Q,[br]then I will be able to calculate

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the P or not Q values and once[br]I've got that I will be able to

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use the value for the event[br]operation. And because there are

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no other if their operations.

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I can use the even operation[br]here instead of having to write

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the whole thing out because then[br]that would again just take up

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too much space in here.

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So let's get started P&R I'm[br]using these two columns, the P

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and they are columns and I'm[br]using an end again. What do I

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know about the end and is only[br]true if both of them are true?

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This is true for both of them.[br]That is not so it will be

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forced. That's true for both of[br]them. That's going to give me

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true that is not gives me a[br]force that is not give me a

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force force force and force

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again. Then I'm going to have to[br]use the Q&R so Q&R both of them

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are true, gives me back a true[br]value. They are different. That

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gives me back of force value.

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They bought force that still[br]gives me a force value because

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it's the end. Both through gives[br]me true Force force force. Now

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that's great. What I need to do[br]next is to find out what is the

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overall value combinations in[br]the square bracket. Now in this

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work square bracket I'm[br]combining together these two

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using the OR. So this is this[br]column and that is this column

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and I'm going to need to combine[br]them together using the OR. What

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do I know about the?

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Or the OR is only force if both[br]of them are forced in every

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other case, is the OR returns[br]true? So that's true. This is

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force. That is true. That[br]is force true Force, force

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force. So I've got the[br]square bracket.

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Next thing I need is[br]the note QQ is here.

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So they're not. Q would be[br]just where Q is true, not

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QB LB force.

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An when Q is force not not Q[br]will be true.

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I've got the not key now and[br]what I need is the P or not Q.

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P or not, QP is this one.

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And not Q is this one that they[br]are quite far away from each

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other, but it just an or[br]operation. Again, if at least

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one of them is true, then the[br]orbital return are true. So true

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force gives me true. True force[br]gives me true true. True gives

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me 222 gives me true force and[br]force gives me a force force and

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force give me a force for so[br]true gives me a true and false

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or true. Me too again.

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And then the last step.

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Is that I need to use?

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The If then operation.

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Which two columns do I need to[br]use the Ethan operation for? I

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need to use for this?

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And I need to use for the[br]square bracket. Now this is

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my a because this is what on[br]the left side for the event

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symbol and this is my B[br]because this is on the right

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side of the if then symbol.

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True if then true returns a true

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value. Force even true returns a[br]true value true if then two

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returns at true Value Force if[br]then true returns with true

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value. True if then force[br]returns a force, why you force

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if then forced returns a true[br]value force if then true returns

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a true value, and force Tipton[br]true again returns 2 value.

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There is a curiosity about this[br]to table because if you look at

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the final output values, there[br]is only 1/4 output in here.

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Everything else is true.

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Let's finish this video with a[br]very complicated truth value

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table. Let me show you what you[br]have to do. If you have gold 4

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input values, so.

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PX or Q.

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And not R.

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If and only if.

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Not Q or S.

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Now I have got four different[br]letters PQ R&S not 2 to 4 =

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16. So I'm going to have a[br]rather long truth value table.

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So let's start building it up.[br]I've got P.

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I've got Q.

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I've got our.

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And I've got ***.

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Now what's the half of 16[br]half of 16 is 8, so I'm

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going to have to start with[br]putting down 8 truth.

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And eight forces.

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Just about then the[br]next step I'm going to

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put down four truths.

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N 4 forces.

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4 two again.

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And for force again.

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Next, 1/2 of four is 2, so[br]two through 2 fours.

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To to force force again until[br]I finish with the whole table.

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And in the last one is just 3/4,[br]alternating all the way down.

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OK, does this start up the[br]table? Then what I need to

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look at is breaking up the[br]brackets, finding out the

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priorities so there is a[br]square bracket and within the

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square bracket I have got a[br]normal bracket and are not so

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I can cut combine together PX[br]or Q.

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Then I would need to[br]find out the note R.

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In the five, being able to[br]calculate that, I will be

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able to calculate what is[br]inside the square bracket.

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Then once the square bracket has[br]been done, then I need to look

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at what's on the other side of[br]the if and only if symbol on the

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other side. I have got not Q or[br]S to be able to calculate that.

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First I need to calculate not Q.

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Then I can do the not Q or S.

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An once that's been done, I can[br]calculate the if and only if,

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which will give me the final[br]output. So let's get started PX

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or Q. What do I know about the X[br]or the X Rays, the xclusive, or

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which is only true if one or the[br]other is exclusively true? So

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basically, if the inputs are[br]different than the X, or will be

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true PX or Q?

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So the first 2 columns[br]they are the same, so the

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actual returns are force.

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They are different. The EXO[br]returns are true.

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They are different, so the[br]exhaust air returns are true.

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And in here they are the same,[br]so the X or returns are force.

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So the next column in R2 value[br]table will be the not R. So

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where are is true, North Star[br]will be force an when R is

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force, not R will be true.

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Force Force 3[br]Three Force Force

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3. Two force for street due[br]again, so I've got my starting

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values again. I need to[br]calculate what's inside the

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square bracket, so what's inside[br]the square bracket I'm combining

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together the X or and the North[br]R using the end operation.

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What do you remember about[br]the end operation? It's only

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true if both of them are[br]true, so let's came through.

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That is, both of them are true,[br]so I'm going to place a true

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sign there, same here.

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Here.

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Here.

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And the rest of them are all[br]different or double forces, so I

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can just fill in the last of the[br]table with force.

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Then calculate not Q while Q is[br]here and not Q will be just the

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opposite of what Cindy Q column[br]so true will be force.

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Annefors will be true.

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And then what I need to do here[br]is combined together, not Q.

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An S.

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Using the OR operation. So[br]what do I remember about the

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OR the OR is only force if[br]both of them are force? So

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let's try to filter out[br]those inputs, not both.

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Force both of them are[br]force, so this will be a

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force in here.

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Both of them are force,[br]so that's another force.

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Both of them are force.[br]Here is another force.

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Here is one more.

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So the rest of them I[br]can fill in with truth.

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Lovely, so the final answer is[br]the if and only if and what do I

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combine together to get the[br]final answer? On one side I've

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got the square bracket which is[br]here and the other side I have

0:14:51.083,0:14:56.888
called the note Q or S which is[br]here. So if and only if returns

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true. If the inputs are the same[br]and it returns force if the

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inputs are different.

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To enforce difference, so the if[br]and only if gives me force force

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and force the same. So the if[br]and only if. It gives me true.

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Different force, same true,[br]different force, different

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force, same again true,[br]same different same.

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Same.

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Different gives me a force[br]different gives me a force

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different, gives me a force,[br]different, gives me a force and

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a different at the end. Again[br]gives me or force. So this is

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the final answer for this long[br]and rather complicated Boolean

0:15:45.546,0:15:49.794
expression. And as you see there[br]are quite a bit of different

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combinations of force and truth.

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I hope that by now you have[br]been able to build a good

0:15:58.093,0:16:00.901
understanding of the Boolean[br]expression and they threw

0:16:00.901,0:16:04.411
tables. I have a few last[br]practice questions for you

0:16:04.411,0:16:07.921
and the answers will follow,[br]so these are the practice

0:16:07.921,0:16:08.272
questions.

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And here are the answers.