1.1 Logic Part 1 - 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
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Intro to Python here on Cybrary on-demand.
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I am your instructor as always, Joe Perry.
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Today we're going to begin Lesson 1,
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programming basics logic.
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In this lesson, we're going to learn about
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Boolean logic and logical operations.
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Those operations are going to be AND,
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OR and NOT and we're going to learn to
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implement those logical operations in Part 2,
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by learning what the logical laws
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and the derived operations,
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most specifically XOR is going to be
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the one that really focus on.
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What is Boolean logic?
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I've used that term in a couple of these videos already
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and it's important to understand
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what it is and where it comes from.
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Boolean logic was published by a man named
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George Boole in the book and
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the mathematical analysis of logic,
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which is a page turner,
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I assure you. [LAUGHTER] It's not.
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It's an incredibly dry and extremely boring book.
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It was published in 1847 and
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like most things, it's terrible.
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That was mean to say,
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but I'm going to stick to it.
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Stick to my guns. [LAUGHTER] More seriously though,
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is influenced by a Sanskrit philologist, Panini.
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I think that's how you pronounce that name,
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but honestly, my Sanskrit is rusty.
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I haven't taken it since high school.
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You know how it goes?
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Panini was a philologist,
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is a person who is a grammarian.
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They study the evolution in
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the design of languages and grammar and how
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they function and he
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was quite possibly the first person to
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apply formal logic to
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the concept of linguistics and grammar.
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Over the course of creating
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that work and over the course of documenting that,
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Panini developed the concepts
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of truth and falsity that were then
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adopted to create Boolean algebra and Boolean logic.
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There are only two values in Boolean logic.
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It's not like normal math where there's
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an infinite number of
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numbers and arrays and things like that.
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In Boolean algebra and Boolean logic,
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there are only two possible values,
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one and zero,
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and then there are only three possible operations.
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There's AND, which is
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represented in formal logic by a period
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and usually in shorthand or informal or programming,
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it's represented by an ampersand.
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There's OR, which is represented by
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the plus sign or by the pipe.
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That's the straight line, up and down.
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Usually it's going to be on
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the same place as your forward
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slash and then NOT,
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which is usually represented by a tilde.
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So those are the only two values and
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the only three fundamental operations of Boolean logic.
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But from those very simple concepts,
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were actually able to derive all
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of modern computer science,
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all Turing complete machines,
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and a pretty substantial chunk of mathematics.
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Now, there is, for those of you who are really
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excited about the most modern technology,
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quantum computing does play with that a little bit and
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introduces a relativistic and probabilistic states.
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But fortunately, this is an intro to
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Python video and not a quantum computing video.
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So save your hate mail for when I earn it later.
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[NOISE] In this lesson,
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we're going to spend a lot of time on truth tables.
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If you've never had the joy of
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going through a CS101 class,
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a truth table is exactly what you see in front of you.
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It's a table of some number of values and
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statements which is used to
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represent every possible state of the equation.
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In this case we have two inputs,
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A and B, and you'll see those use pretty often.
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With A and B, we represent every possible combination of
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those two Boolean factors
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or Boolean values, true or false.
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By the way, in computer science,
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when I talk about true and false,
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that's the ones and the zeros of binary.
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Binary is an implementation of that Boolean logic.
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We're going to use one or zero instead of
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true and false in most of these places.
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Don't let that trip you up too much.
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So here you can see we have two ones
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and two zeros under A.
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If we just had A in this table,
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it would just be two lines of one and zero.
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But because we have two factors,
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we have to represent every possible
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combination and for those of you
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who know binary and understand how to read binary,
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you know that with two digits you're
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capable of representing four numbers,
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0, 1, 2 and 3.
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In a useful equation when you're building
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your truth tables is however many inputs you have,
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it'll be that power of two number of rows,
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which is to say, well, that power
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of two number of possible combinations.
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So that many rows, which is to say,
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for example, if you have three
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inputs is going to be eight.
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If you have four, it will be 16.
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If you have five, it'll be 32,
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on and on and on and on.
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If you have five inputs,
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I generally don't recommend
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building a single truth table.
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We're not going to deal with any truth tables larger than
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three inputs and two or three statements at a time.
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Your statement is going to have
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the same number of rows as
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your inputs and that's just
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going to be whatever equation you're doing.
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We'll see that it'll make a little more sense,
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I think in the next couple of slides,
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we'll see that play out.
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But your statement is going to be just all of
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the solutions with those inputs.
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As an example here, we have the first logical operation.
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We see that we have true
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and false represented with A and B the same way
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we do in this previous slide with ones and
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zeros and we see that and evaluates to true,
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that is, and evaluates to one if and only if so,
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only in the case that all of
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its input values are also true or one.
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On this table you can see that on our first row,
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A is one and B is one.
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So that's representing the binary number 3.
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The AND, A and B we're going to
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represent here, [NOISE] is one.
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Because of the fact that both of the inputs were true.
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The output of AND is true, pretty straightforward.
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The next row we have A is one which,
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helps us to evaluate this,
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but B is zero.
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Now, I'll say again and evaluates to true
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only in the case that all input values are true.
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Which means that with B being zero,
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A and B is also zero.
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As you can see on the next row,
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A is zero, B is one,
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still have a zero in it and on the final one,
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of course both of them are zero,
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therefore AND is zero.
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A useful shorthand is that generally speaking,
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if you add all of your inputs together in a truth table,
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you're generally only ever going to have
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one row evaluate to true.
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It's not 100 percent the case,
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there are complex inputs that can mess with that.
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But as a basic rule,
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you can generally trust
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that and I want to mention real fast.
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I always skipped it, AND is often referred to as
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the logical conjunction in formal logic classes.
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So if you hear that, that's what that's referencing.
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Now, OR logical disjunction OR is similar to
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AND in that it evaluates to
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true based on whether or not an input is true.
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It's different in the sense that if any of the inputs to
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OR are true OR will evaluate to true.
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So not exactly the inverse of AND,
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to some extent it feels that way and it
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looks that way on the truth table as well.
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For example, on our first row
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we have A and B both equaling one.
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Now, it doesn't matter if both of them are true,
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so long as one of them is true.
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So A or B evaluates to true.
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On the next row, we see that A is one and B is zero.
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Now with, and you'll remember that evaluated to
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zero because one of the inputs was false.
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However, because one of the inputs is true for OR,
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it evaluates to true.
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The next row, same thing with B being
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true instead of A and then the final row,
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because none of its inputs are
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true or will evaluate to false.
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The third and final fundamental logical operation
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of Boolean logic is NOT,
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or the logical inversion.
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NOT evaluates to true if and only if,
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only in the case that it's input is false.
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NAND is generally used for single inputs or it is
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used for the results of an equation.
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For example, you can see here that I have NAND and NOR,
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which are two very commonly used gates in logic.
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Gates and operations are differentiated in
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that a gate is something that
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is actually physically constructable.
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Even though the AND,
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OR and NOT are the fundamental operations,
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NAND and, NOR are the two most commonly
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used fundamental physical gates of logic.
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Don't get wrapped around the axle with that,
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you aren't going to need to learn how to create
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a circuit board or a circuit diagram.
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Don't worry about that at all in this class.
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Point is, NAND and NOR are
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two very commonly used logical operations
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which are derived from the three fundamental.
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So here you can see that NAND,
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which is NOT AND,
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is going to be evaluated based on the input from AND.
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So, AND we already established is
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going to be 1, 0, 0, 0.
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NAND is just the logical inversion of that.
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We have 0, 1, 1, and 1.
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Very straightforward, very easy
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to wrap your head around once you've
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seen it a couple of times,
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A or B, you can probably guess because it's 1,
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1, 1, 0 and NOR is the logical inversion of that.
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We're going to have 0, 0, 0 and 1.
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At the next lesson, we're going to do some math.
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We're going to jump in and really start
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learning about some computational logic.
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We're going to learn about some
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>> fundamental laws of logic.
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>> Here I've included a comic from the inimitable XKCD.
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There are going to be a lot of
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XKCD throughout this course.
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Because Randall Munroe is frankly
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a genius who has made the world a better place.
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That's said before we go onto the next lesson,
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we're going to take a second,
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we're going to do a knowledge check.
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We're going to break these videos upload
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or the next part of this lesson.
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Break these videos up a
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little bit just so you have time to
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digest it and make sure you've learned
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the actual material.
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Which operation is the logical inversion of AND?
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What is derived from the combination of AND and NOT?
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The answer is NAND.
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[NOISE] What are
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the two possible values of Boolean logic?
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There are only two, [NOISE] true and false.
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What are the three fundamental operations
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of Boolean logic?
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There are lots of possible operations.
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There are only three that are fundamental.
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AND logical conjunction,
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OR logical disjunction, NOT logical inversion.
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Summary in this video we cover basically
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what you just checked out on that last slide,
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the two possible values,
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the three possible operations,
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and discuss some of the derivations from those.
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Again, in the next video, we're going to come back.
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We're going to learn about the laws of
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logic or we're going to apply them.
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Until then, I am as
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always thrilled to have you here and
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hope to see you back in our next video.
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I'm your instructor Joe Perry,
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and this has been intro to Python.
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