1.2 Logic Part 2 - IP

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Time
2 hours 57 minutes
Difficulty
Beginner
CEU/CPE
3
Video Transcription
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>> Hello everyone, and welcome back to intro to
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Python here on Cybrary On Demand.
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As always, I'm your instructor, Joe Perry,
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and today we're here talking
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about the programming basics,
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specifically focusing on logic.
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This is part 2 of Lesson 1.
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In part 2, we're going to be discussing
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the laws of logic.
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Here's our first one, the law of idempotence.
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Now the law of idempotence or the law of idempotence,
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depending on the accent of
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the person who taught it to you,
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is very similar to the law of identity in arithmetic.
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Basically, all it's saying is that when you
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perform an operation with only one input,
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A or A, A and A,
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the output is just going to be whatever the input was.
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The obvious exception from that is
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the inverse operation or the logical inversion operation,
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which will flip the input.
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But with Or and And,
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it will just be whatever the input was.
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This may seem obvious,
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but it is actually one of the fundamental parts
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of Boolean logic,
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the way that one equals one
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is actually a fundamental proof of mathematics.
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The law of association.
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The law of association is a trickier one.
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I don't really like any of
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the verbal definitions that are given,
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which is why I just use the equations in
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a truth table here to help you understand it.
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Basically what the law of association
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is that if all other things are equal,
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if you're performing the same operation
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with all of your inputs,
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it doesn't matter the order in which it's done.
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Normally order matters,
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and in logic,
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you're going to do it very similarly with mathematics.
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Inversion is going to come first,
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and then you're going to do whatever is in parentheses.
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You're going to do whatever is in
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parentheses and then inversion.
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Parentheses will always come first,
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of course, with mathematics.
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Here you can see this is just an easy demonstration
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that nothing changes in these types
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of equations when you move the parentheses
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around because all other things are equal.
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The law of commutation is simply
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saying that the order of operations,
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the order of inputs to a single operation don't matter.
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A and B is the same as B and A.
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You don't really have the case
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like with subtraction in math,
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where if the order is off,
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you're going to get a different answer.
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It doesn't matter about the order with
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a single operation.
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The law of distribution is a pretty interesting law here.
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What it's saying is that if you perform an operation
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against the results of a sub-function,
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something like B or C in parenthesis,
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performing and against that is the same as
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performing and against both of
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those separate values and/or in the result of that.
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That's a tricky thing to understand.
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Here I have this truth table built out.
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The law of distribution you're not going
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to use terribly often.
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A lot of these logical laws you're really
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going to use just conceptually,
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not really applying them very rigorously.
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But having this truth table here is very helpful because
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it helps you understand the way the laws work,
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even if trying to come up with
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a good verbal definition is challenging.
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You can see A and B or C is going to evaluate to 1110,
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and then the rest are all zeros.
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Then the same way with A and B or A and C,
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it's going to evaluate to the exact same thing.
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That's like I said,
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that's the great thing about truth tables.
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I know a lot of people don't really like them
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because of bad experiences.
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When you're using a truth table you can put all
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of the possible inputs and outputs in front of you,
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and really get a good sense of
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what the equation is and how it works,
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and be able to identify relationships
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you would otherwise potentially miss.
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The last law we're going to talk about is
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a law called De Morgan's law.
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It's my personal favorite because I
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think it's absolutely fascinating.
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The idea of De Morgan's law is,
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the inverse of A and B equals inverse A or inverse B.
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As well, the inverse of A or
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B equals inverse A and inverse B.
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You can see that on the truth table.
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I find this absolutely fascinating personally,
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and hopefully, you find it interesting as well.
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The idea that because
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of the structure of logical inversion,
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and I alluded in the first video that Or and And have
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a theoretical inversion in
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terms of the number of true values they have.
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Here you can see that when you break it out,
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it's represented again on the truth table.
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Even if it seems weird or hard to understand,
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you can see that it just is
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true by examining the inputs and the outputs here.
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Now there's a third operation
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that I want to talk about here,
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not a third operation, but a new operation
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rather called XOR.
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XOR is the concept of logical exclusion,
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which is to say one
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and only one of the inputs can be true.
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Or as you might hear it more often,
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the inputs must be different.
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If A is 1, B must be 0.
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If B is 1, A must be 0.
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Now, XOR is extremely important to computing.
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In fact, XOR physical gates
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are the most common gates along with NAND.
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XOR is used for encryption and
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all sorts of other programmatic functions.
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It's actually fascinating all the places
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that XOR can show up.
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But that operation is not actually
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its own fundamental operation.
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As I said in the last video, there are only the three.
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Instead, XOR is a derivation
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of the three operations combined.
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Now what I want to do now is I want to give you a chance
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to try and figure out what that operation is.
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I'm just going to pause. I'm going to give you
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the chance to pause the video here,
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so go ahead and pause.
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Either you figured it out or you've come back to
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find the answer, and it looks like that.
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Now that obviously is a pretty complex equation.
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It's more complex than anything we've looked at so far,
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A or B and the inverse of A and B.
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The reason why I put this up here is just because I
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think it's absolutely fascinating.
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With just those three gates,
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you're able to derive a completely different process.
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Not only derive a completely different process,
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but one of the processes that makes computing possible.
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All of computer science is born
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just from these three operations, these two values.
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When I say all of computing,
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I mean everything that you're
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currently doing on your screen,
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the video you're watching,
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the images on it,
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all of me talking into this camera and being recorded,
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all of that is a product of
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just two values of three operations.
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That is absolutely fantastic and mind-blowing to me.
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If it's not to you, I'm sorry that
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the joy and excitement is gone from your life,
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but it's super cool,
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and you should take a
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second and consider how fascinating that is.
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In that last slide,
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I showed you a complex operation.
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In this one, I want to give you an idea.
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This slide, and the next one, I'm going to
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show you how to break that down,
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how to think about logic problems like this.
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It's really important to take
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a big problem and break it down into its component parts.
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You're going to see that a lot
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throughout the course of this video.
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Throughout the course of this lesson and
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this actual course and the next two Python courses,
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you're really going to want to be able to
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break problems down into their component parts.
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It's just a fundamental thing that you want to know
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how to do in computer science.
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Here you can see that I have this operation,
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inverse A or B,
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in parentheses, and inverse A and B.
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You can see here that instead of trying to do
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that very difficult complex equation,
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I just broke it up into pieces.
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I said, inverse A or B can be turned into its own input,
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which is going to be called N,
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and that we will derive from the previous values.
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We're also going to put an inverse A in
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its own column just to make it easier to use.
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Also with M, we're going to do inverse A and B.
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Then in the last column is just going to be
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ending those two columns together.
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We're not going to worry about trying to
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remember which value we put where and who's doing what.
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We're going to break this into its constituent pieces
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and solve them individually.
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You can see here that that's going to make
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the process just so much easier.
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That's true not only in this class,
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not only in programming, but really in life.
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Finding a way to break down your problems into
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their component pieces, their simplest,
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most fundamental level, solve them individually,
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and then use that to solve the whole.
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Knowledge check; what do we go over in this video?
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The following are examples of which laws?
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A and B or C is the same as A and B,
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or A and C. That's distribution.
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A or A equals A,
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that's the law of idempotents or identity is
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generally what I refer to it as.
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A and B equals B and A,
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that's the law of commutation.
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Of course, A or B in parentheses,
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or C is the same as moving the parentheses around,
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which is the law of association.
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Which super common logical gate
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is derived from the three core operations?
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If you remember from before I
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rant about how cool computers are,
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you'll remember that that is XOR.
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In this video, we talked about the laws of
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logic, derived operations,
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and I break down complex problems
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into their constituent pieces.
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Hopefully, you learned a little bit, and hopefully,
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it's inspired you to come back for the next lesson,
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Lesson 2, which is going to be programming basics.
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We're going to learn about how to
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use variables in programming.
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I, as always, I'm your instructor,
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Joe Perry and as always,
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I'm very excited to have you here,
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and I hope to see you in our next class.
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