 Page 76, please. So we did a couple of days looking at deductive reasoning, the ability to make a conclusion, a logical conclusion based on some true statements. And I'll start out, first of all, any questions from number one and number two? Remember I said to you that number three and number four, I would sort of go over in class, although I didn't assign number four. By the way, one student did actually solve number four yesterday in class, pointed out to me and did a very lovely job. So well played. Some of you are really getting this. I found, Alex, I found proofs tough. When I got to college, that if you asked me, what did I struggle the most in in math? A lot of the stuff I found easy. Proofs I found very difficult. There's a science to them and there's an art to them. And by my third year in university, I had to hang in them. But my first couple of years, it was kind of hit and miss. So if you're finding these weird, you're normal. Hopefully you found number one okay though. You're able to, if you give me two statements, come to a logical premise or a logical conclusion. But are there any questions about number one and number two? Yes? Yep. Let's see. So what do you get as an answer in choice one, choice two, and choice three? You get the same answer every time or what do you get? What do you get? Six each time? So the conjecture is any number I stick in less than ten will work. Maybe any number at all will work. Prove it. I'll do a generic number N. You ready? Add four, right? Double it. Two N plus eight. I doubled everything. Add seven. Two N plus eight plus seven is two N plus 15. I have to zoom in here so I can write in this small blank. I can't fit this all on one screen at this magnification so I'm going to do mine in choice three. N plus four, two N plus eight, two N plus 15. Add on the original number. What was my original number here, Joe? N. So if I add N to this instead of two N's, you know how many N's I'll have? Three N's. Three N's plus 15. See where we're going? I don't know where we're going to end up but I can follow this through. Let's see. Divide by three. So if I divide by three, I would divide that by three and that by three. What's three N divided by three? What's 15 divided by three, okay? Subtract the original number. What was my original number? So if I subtract N from this, you know what's left behind? Just the five. Add on the number of the month that you were born, okay? Let's find out if the month that we're born will keep track of what happens to it and we'll see if it just suddenly vanishes later on anyways. So what month were you born? What number? Six. So that would make this, I'm going to write this as five plus six. For me it'd be five plus seven and I'm going to see if your six just vanishes later on. And then the five is continue dropping down. Add on four, nine plus six. I don't want to touch that six because that's your month. Times by a hundred, nine hundred plus six hundred. Add on the number of the day that you were born. What? Three. So I'm going to write this as nine hundred plus six hundred plus three. That came from the month. That came from the day. So mine would be a seven hundred plus thirty one. Times everything by a hundred, okay, nine thousand plus six thousand, no, times by a hundred Mr. Dewick, ninety thousand plus sixty thousand plus three hundred, yes? Add the last two digits of the year that you were born. So add the last two digits of the year that you were born, what year were you born? Nineteen ninety five. That's going to give me a ninety thousand plus a sixty thousand plus a three hundred plus a ninety five. And when you subtract ninety thousand, you're going to get this value now. I'm going to guess for you, there's your month, there's your day, and there's your year. It's your birthday. Is it not? You were born June third, nineteen ninety five. This will cancel, right, and I'll be left with that, that and that, which is why you got sixty thousand, three hundred ninety five. See I think everybody who does this, actually their answer is going to be their birthday. No, no, no, there's a ninety thousand plus sixty thousand plus three hundred plus ninety five. If I subtract ninety thousand, all that's going to happen is this is going to vanish, but I'll have a sixty thousand and a three hundred and a ninety five dropping down, which is your birthday. If I had done this for me, sorry, the other answer is all would have been sixty thousand three hundred ninety five for you. Yes or no? You told me you got six zero three nine five for all of them. So isn't that six zero three, if you add that together, isn't that six zero three nine five? Right, I'm saying that, that's what you got, and now I can see where that six came from. You see how your month dropped down and we ended up moving it right to there? My month would have been a seven. It would have dropped down, it would have been a seven there, a seven there, a seven there, a seven there, a seven. I would have ended up with a seven in front, July. My day is thirty first. I was born on July thirty first, so I would have put a thirty one right there, a thirty one, a three hundred and ten, sorry, a thirty one hundred right there, a thirty one, I would have ended up with a thirty one sitting right there. And the year that I was born, nineteen sixty nine, that would have dropped down there too. So I can kind of see how this one works. I think all of you should have got your own birthday, and you know what? I don't think it has to be a number between one and ten. It looks like this would work for any number, pick any number at all, and eventually you'll end up with your birthday as long as you plug in your numbers. This is a fairly tough one by the way. This is harder than I'd feel comfy asking on a test. Is that okay? They agree? Any others? So let's look at number three real quick. Number three says, try and prove these ones. Well, for consecutive even numbers, I would call them two n, two n plus two, and two n plus four. Those are three even numbers. How do I know they're consecutive, Joe? Because here's your first one, two more, and four more. That's how you'd get your next two. That's what I would start out doing. What does the word some mean, Joe, times? What does some mean? You need to know your math words. Some means add. So the sum is divisible by six. If I add, I get what's two n plus two n plus two n, six n. What's two plus four gathering like terms? Does six go into that? How do you know six has to go into it? Does six go into there? How do you know? There's a six in front of everything. Does six go into there? Yeah, I know six goes into six. You know what? I guess if you add any three even numbers that are in a row, the answer will be divisible by six. There's a proof. Any of the other ones you want me to try, are you wondering about? I'll do a couple of more of these later, but I've been talking enough I want to get to today's lesson. Turn the page. This one I did not type out because I thought the textbook didn't do a brutal job. Can you go to lesson three? The heading is, and this might be a new word for all of you so you can underline it, we're going to investigate a fallacious proof. Fallacious comes from the word fallacy. If someone says there's a fallacy in your argument, what does that mean? Nobody know what the word fallacy means. Sorry. You've made an illegal chain of reasoning somewhere along the way. There's a fallacy in your argument, somewhere in your logical deductive reasoning, two of your steps don't go together. What we're going to look at is trying to analyze the validity of an argument. Here's a very, very famous one. You can tell that we've done something wrong in this argument because we're going to end at a stupid conclusion, a nonsense conclusion, which means somewhere along the way in our chain of reasoning, we have a fallacy. Here's the proof. They start out saying, let A equal B, here's the setup. This is an equation. We're going to add A to each side, so I'm going to go plus A plus A. Is that allowed? Are you allowed to add the same thing to both sides of an equation? Yeah, math eight. What is A plus A? What's an easier way to write A plus A? Two A. Here, for this step, Joe, I'm going to write gather like terms. That's where this line came from. Liam, can you read step three of this line to me? Out loud, read it out loud. Read this line to me. Now read step four to me, please. What did they do to both sides in step four? Can you see it? That's why I wanted you to read them both out so we could see the difference. They've gone minus two B from each side. Is that illegal? Are you allowed to do the same thing to both sides of an equation? I think I'm okay with this. Shae, can you read step five to me, please? Stop. That looks different from above. How the heck did they get this from that? What did they do here? You've got to remember your math 10 and math nine. How did they turn two A minus two B into two bracket A minus B? What did you call that last year? It begins with a letter F. They factored. What did they factor? GCF. Remember doing that last year? Here they've gone GCF. Shae, I didn't have you read the other side because the other side didn't seem like it had changed all that. Well, wait a minute. What did the A plus B come from minus two B? How did we get an A minus B over here? Devin, can you see it? They went positive B. Take away two B. That's where the negative B came from. They also did like terms. Bender, is that okay? So a little weird. I know. What are they doing here to both sides? You see it? What are they doing to both sides here? I heard someone say dividing, yeah. Dividing what? Marcus, are you allowed to divide both sides of an equation by a number? Well, yeah. It's how you got the x by itself. You would divide by five, divide by five, or divide by ten, you get the x by itself. Marcus, what is A minus B divided by A minus B? What's anything divided by itself? Five divided by five is zero. Eight divided by eight is? What's A minus B divided by A minus B? That's why they wrote a one here. What's A minus B divided by A minus B? That's why they wrote a one here. What is two times one? So we end up with two equals one. Is that true? Does two equal one? That must mean there's a fallacy in our argument somewhere because I've ended up with a conclusion that I know has to be false or I'm in trouble. All the math I've learned so far is wrong. Real question is then, where is the fallacy? Where did I zig it? Where did I make my mistake? Let me see it. Yeah. Emily says step six, and believe it or not, Emily's right. Why step six? So Emily's thought is it has something to do with the negatives. She's very, very close, but I've got to be really fussy. There's a key reason why step six is illegal, and it has to do with the very first line, step one. Aaron, can you read step one to me, please? Read it nice and loud again without the word let. A equals B. What did you just say? Okay, that's what we started out. That was our initial premise. So look at that bracket in step six. What's A minus B if A equals B? What is A minus B? Emily, what you're really doing here, because you said that, you're dividing by zero. That's a huge no-no. We've said for years you can't divide by zero, and this is one of the reasons why we don't allow that. If we allowed you to divide by zero, you get stupid math, dumb math, bizarre math, yucky math. Where is the error? Step six, why, Joe? If A equals B, then A minus B is zero, because that's the same as going A take away A or B take away B. If they're each the same number, you're going something minus itself. We can't divide by zero. It's all caps, so that should be shut. We can't divide by zero. Are you getting stale? I need to actually go to the dollar store and get some prizes, maybe. I can quite figure out a way to do this. The fallacy in this proof here is without realizing it, because you said A was the same as B, when you divide by A minus B, you're dividing by zero, and as soon as you do that, that's a problem. Why can't you divide by zero? Because of this. Other reasons as well. Put your pencils down. Look up. I can't remember if I've done this with you or not. If I haven't, I am now. Benedict, put six divided by three. That also means that two times three is six. If you went six divided by zero, and you got an answer, I'll call that answer X. That would also mean that X times zero was six. What's wrong with this line? What's anything times zero? Can you ever go X times something equals six? This is actually saying, there's no possible answer that will work, because for this answer to work, it would have to be able to go backwards, and nothing does. There's also a fallacy in that argument right there. Page 80. Here is your vocabulary words. Fallacy, fallacious if you want the noun, and validity. In mathematics, a proof is valid if the reasoning is true in every single step of the proof. We say that the proof on the previous page is invalid, because one of the steps, step six, of the reasoning is not correct. It is correct to divide both sides of an equation by the same quantity, except you can't divide by zero. In step one, we said that A equal B, so A minus B is zero. Even though all the other steps are correct, all it takes is one single improper argument to reduce the whole proof in valid to make the whole, I'm terrible. Even though all the other steps are correct, it only takes one improper action, one improper step to make a proof invalid. Proofs are like houses of cards. You ever tried building a house out of a deck of cards? You move one piece and the whole thing comes crumbling down, which is why proofs are tough. It says this, in mathematics, an argument is when two or more statements or propositions called premises or conjectures are used together to form a conclusion, an argument can be valid or invalid, as in the next value, the example. It is possible to have a valid argument that's not true. It is possible that in your argument you haven't broken any logic rules, but you still ended up at a conclusion that might be incorrect. Let's get to example one. Consider the following two arguments. Argument one, all women are more whole. Argument two, statement two, Anne Irwin Young is a woman. Therefore Anne Irwin Young is more whole. That's our first argument. Argument two says this, some people who cough have the flu, Jaden has a cough, therefore Jaden has a flu. Which one of those arguments has an error in the reasoning? Which one of those is a fallacious proof has a fallacy? Yeah, Taylor, why? So some does not imply all. Argument two, some is not the same as, that's my abbreviation for is not the same as, is not equal to all. Can you cough and not have the flu? Yes, based on those statements. Thank you for coughing. We do this all the time in society, getting some and all mixed up. Racists do it all the time. They try and stereotype an entire race based on one person's actions. Politicians do it an awful lot. We do this all the time. Bad math, bad arguing, bad logic. Argument eight, age 81, the following is a famous story in logical reasoning. Ignore the first paragraph, three people want to stay at a hotel. They arrive late at night, second paragraph, and they're very tired. The hotel has one room left, but it's still under renovation. The manager gives them a discounted rate since the room is not completed. And they are attending the conference. So here's what the manager charges them. 30 bucks for the room. The three friends contribute 10 bucks each. How much money have they paid? 30 bucks. The next morning, the manager rethinks the rate and decides to give them a further discount, hoping that they'll book again the next day. The manager gives the bellboy 5 bucks to take to the room and give back to the three friends. The bellboy realizes the three fans aren't expecting any sort of refund, and he feels it's going to be too difficult to split 5 bucks equally. He decides to give them back only 3 dollars, and he keeps 2 bucks for himself. So he gives the three friends the 3 dollar refund. Each friend gets a dollar. The three friends had originally each paid 10 dollars, but each of them received a dollar back. Now they have only paid 9 dollars each for the room. They are happy that the manager has given them a discount, and they'll come back to the hotel next year, and the bellboy is happy because he's got an extra 2 bucks in his pocket. Bill, a student, reads the story above, and he says the total cost of the room should now be 9 dollars per person times 3 people, plus the 2 bucks the bellboy kept 29 dollars. Where'd the extra dollar go? How many of you heard something like this one before? None of you have! Oh man, my elementary school teacher gave this one to me. We started out with 30 bucks. At the end here I have 29 bucks. Apparently. Or do I? Did we lose a dollar? How much did each person pay for the room? 9 bucks? How much did the bellboy keep? 9 times 3 plus 2 is 29. Where'd the extra dollar go? What do you think about this one? Boston, you awake? Here's your homework. Try number one. Write the reasoning used in each step if you can. If you're not sure, you can take a peek at the back. And then we end up with 2 equals 1, which I know is incorrect. I mean somewhere here we made a faulty line of reasoning. Somewhere here there's a fallacy. Try number two. You end up with 5 equaling 4. Does 5 equal 4? Then somewhere along the way we've made a mistake in our reasoning. See if you can figure out where. Try number five. Number seven. So in number seven they're giving you four arguments. Argument one, argument two, argument three, and argument four. Three of them have an error in reasoning. See if you can spot which only one of those is a legitimate argument. And number eight.