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Well for assignment 6 I'm just going to do the last 3 questions, number 7, 8 and 9.

Â I think you should be able to do the first 6 on your own and check it on your own or

Â with other students. So in number 7 we have to negate

Â statements and put them in positive form. Part A was for all x in N, there was a y

Â in N X plus Y equals 1. The negation for that is, and I think

Â you'll get the same answer as I do, if you get it right.

Â There is an X and N, such that for all Y and N, X plus Y is not equal to 1.

Â You, you could have written that last part as not X plus Y equals one if you like,

Â that's equally correct, I just choose this way of writing it.

Â Okay, and in this one, for "X" that's greater than zero, there's a "Y" that's

Â less than zero, "x" plus "y" equals zero, the negation of that, in positive form, is

Â an "X" greater than zero and it's greater than zero, okay.

Â For all "Y" less than zero, these don't change around.

Â They stay the same way, because these simply tell us what Xs and Ys we're

Â looking at. And, we get X plus Y not equal to 0 here.

Â Okay, for part C, there is an X, greater, there is an X, so epsilon greater than 0,

Â negative epsilon less than X, less than epsilon.

Â And when you negate that you get for all x epsilon greater than 0.

Â But in this case you going to have to sort of split this into 2.

Â One of 2 effects is not between negative epsilon and plus epsilon then either x is

Â less than or equal to negative epsilon or x is greater or equal to epsilon I didn't

Â put parenthesis around either of these,[UNKNOWN] I could have done if I

Â wanted to be ultra clear but this takes precedents over the logical operations,

Â arithmetic's expressions are, inequalities and so forth.

Â Take precedents over logical connectors. Because these are connectors.

Â They connect Statements about mathematics. And this is a statement about mathematics,

Â and that's a statement about mathematics. But if you wanted, you could have put

Â parentheses around that. And you could have put parentheses around

Â that. That would been, that would have been

Â another way of doing it. Moving onto the last part.

Â For all x and n, for all y and n, there a z and n .

Â X plus Y equals z squared, we're not talking about whether these or true or

Â not, we're just writing them down and negating them, so you got exists an X and

Â N, exists a Y and N, for all Z and N, and I've just written this again, X plus Y is

Â not equal to Z squared, you could simply put a negation sign in front of that

Â expression. Okay, well that's number seven.

Â It';s fairly straightforward, let's move on to number eight.

Â Question eight is our old friend Abraham Lincoln, and this famous statement of it,

Â or allegedly famous statement of his, and when it's negated[INAUDIBLE] that famous

Â sentence. So, let's let fxt mean you can fool person

Â p at time t. And then his statement is this one.

Â You can fool all of the people some of the time.

Â There were some times when you could fool everybody.

Â You can fool some of the people all of the time.

Â So, there are some people that can be fooled for all ti-, at all times, but you

Â cannot fool all of the people all of the time.

Â Okay, let's just mechanically go through and negate that.

Â Looking at the formulas then, and then we'll try to interpret, the answer and

Â express the answer in terms of English. Okay.

Â So, negate exists, it becomes for all when it for all, it becomes exists, FPT becomes

Â not FPT. Okay, conjunction becomes a disjunction.

Â Again, I'm not putting these in parantheses.

Â And I'm not going to do it here because this is a, this is a, is a whole that

Â quantifies binding the, the very tight binding.

Â And then we have this junction conjunction here and then this junction's here the,

Â their less tightly binding. So looking at this one, exist becomes for

Â all, for all becomes exist we get negation.

Â Conjunction becomes disjunction and the negation here just disappears.

Â So I put a positive statement. Okay so in terms of the formalism, this is

Â fairly routine. In fact, this is such a routine thing that

Â this is essentially algorithmic. I just went through and applied, the, the

Â patterns, the rules that I've observed happening.

Â The interesting part of this, I think, and you wont have to do it, but let's do this.

Â Let's see how we can express this, in English language.

Â And, We're not going to get something quite as, as nice as this, I don't think.

Â Because we need to try and be, to avoid ambiguity as much as possible.

Â So the first part would be the best I can think of at the moment is, let's see.

Â At anytime, there is someone you can't fool, and, oops or, because we've negated,

Â or okay. Let's have a look at this one.

Â I can't really think of anything better than to say the following: for every

Â person, you can't always fool them. Well, no.

Â I think if you try and swap it 'round. You run into sort of a American melanoma

Â type problem. So, to try and avoid that, I think I

Â would, I would want to write it this way. And then, the final clause.

Â But that's easy, of course, let's just say, you can fool all the people all the

Â time. Ok.

Â I'm, I'm moderately happy with that, you might say I have different opinions on the

Â best way to write that. [inaudible], But, number 2 is a bit ugly.

Â But I was, I'm not sure we can make a better job of that one.

Â Okay, in any case, we've, we've, we've, we've negated the thing.

Â And, and this was nice and clean. Okay, now, now, let's look, look at number

Â 9. Well, number 9 involves one of the most,

Â important and most famous, some might say infamous formulas.

Â Of, advanced mathematics. Of university level mathematics.

Â It's this definition. When students entering math-, entering

Â university to study mathematics. Therefore, a math major.

Â It's this definition that, that usually causes them the most problems during their

Â first year. In fact I think it's this definition that

Â probably is responsible for more math majors giving up uh,mathematics in their

Â first year at university more than anything else.

Â It's a really tricky thing to understand as many of you've noticed, and I've seen

Â from discussions on the forum. Figuring this out is, is really hard.

Â This was hundreds of years of effort starting with the invention of the

Â calculus by Newton and Leibniz in the, in the 17th century.

Â It took a long time in several hundred years before mathematicians were able to

Â figure out the notion of continuity and come up with this definition.

Â This was late-19th century that this was done.

Â And it was it was a tricky thing. Negating it is relatively straight

Â forward, actually, because we've, we've got rules for doing them.

Â And so, when you negate it, what you get is that the for all becomes exists and

Â the, it's still a greater than, we're still talking about positive numbers,

Â epsilon. The exists becomes a for all, the for all

Â becomes an exists and, then there's a bunch of stuff that's in the bracket here.

Â This is a conditional, an implication, and so, when you negate it, you get the

Â antecedent conjoined with the negation of the consequent.

Â So, here's the antecedent. Again, I haven't put parentheses around

Â that because this is a, a mathematical statement, and That, that's take

Â precedence over the, over the conjuncts not just junction is I've just connected

Â there operators that bring things together.

Â So that's a piece on its own. That guy, conjoined with the negation of

Â this guy. Okay, the negation of it will be less than

Â F1. If it's originally F1.

Â Okay, so, so that's the negation and that's relatively mechanical to do that,

Â as long as you sort of pay attention to Preserving things that need to be

Â preserved. Changing for alls to exist and negating a

Â conditional. The interesting question is, what on earth

Â does this mean? You know you can read it through as I just

Â did for all epsilon, greater than 0, this is delta greater than 0, such that for all

Â x, yada, yada, yada. What does it, what does it mean?

Â Well, this is capturing in sym-, in, in, in a symbolic language.

Â In an aglebraic form-, formalism. It's capturing something geometric.

Â So, let's see what it's capturing. Let's look at the original definition of

Â continuity. It's about functions.

Â So let's look at a function this way. I'll draw a wavy line vertically, this is

Â a real line, and then I'm going to draw the real line vertically here.

Â So this is the real line, instead of writing horizontally as we usually do, I'm

Â going to write it horizontally. And the function, f, is going to take

Â Numbers here, to numbers here. Okay, so somewhere here we've got A.

Â F applies to that. And it gives me F of A.

Â Okay. Now we're trying to capture the notion of

Â continuity at a. That means when we go slightly to the left

Â or right of a, left or right is up and down the way I represent it, then the

Â numbers don't sort of have a discontinuity.

Â And the way to caption that it turns out and chances are very, very high that

Â you're not going to follow this the first time, it's going to take you weeks if you

Â need to, to master this. But here's what this formula says, it says

Â the epsilons are going to work on here, it says let's take an epsilon here and let's

Â look at F of a plus epsilon and f of a minus epsilon.

Â So I'm going to take an epsilon interval around f of a.

Â And what this definition says, is that, given an epsilon in 1 of these intervals,

Â I can find a delta. So here's a plus delta, and a minus delta.

Â So starting with an epsilon, which gives me an interval here, through any one of

Â those, I can find a delta, which gives me an interval here.

Â Search that. Now let's look at this.

Â Any x, in this region, because this says, that x is within delta of a.

Â So any x in this interval, gets sent to an image f of x, in here.

Â So it's saying, in order to make sure that all the values of the function Are in this

Â interval. I can find an interval around A, such that

Â they're all sent into here. So, if I want to hit the target, imagine

Â this is hitting the target, like throwing darts at a dart board.

Â If I want to hit the target within a specified accuracy of a, of f of a, I can

Â always do it by starting out within an interval of a.

Â So to get within a given interval around f of a, I can always find an interval around

Â a that does it. So everything from here gets sent into

Â here. And if you think about it long enough,

Â you'll realize that what that means is that the function is continuous at here,

Â there's no, no, no jumps. And to try and understand that, let's look

Â at what this guy means in terms of a diagram.

Â I'll do the same thing again, I'll draw the rail line.

Â And I'll draw the real line. And here's a, and here's f of a.

Â Now the negation says there is some epsilon.

Â In, in the previous case this was happening for all epsilons.

Â You could find a delta. In this case there's a fixed epsilon that

Â we can find. And we look at the interval around there,

Â F of A minus Epsilon, okay, what it says, is that there is an Epsilon, such that, no

Â matter what you take here, no matter which one you take here, here we found one of

Â these guys, we said Take any interval here, we could find and interval here.

Â Here we're saying the reason interval here for some epsilon such as no matter what

Â delta we take here, no matter how small you make this delta, no matter how close

Â you get here, that was a point that gets sent to that, you can always find a point

Â in here, that gets sent outside of there. Maybe sent that way, maybe sent that way.

Â So the sum epsilon here, that no matter what you do in here, no matter how close

Â you are to a, something gets sent out here.

Â In other words, there are points really, really, really close to a that get sent

Â outside this region. So there's no way that you can, you can

Â get all points in here. A gets sent to that, but arbitrarily close

Â to a There are points that get sent away. So there's discontinuity, because only a

Â gets sent close to a. When you get, well some other points

Â maybe, but no matter how close you get to a, you'll find points that are sent

Â further away. So that's a discontinuity.

Â Thing's jump, there's a sort of jump form there from there to thing here or here.

Â Now I, the chances are high, if it's the first time you've seen this, that

Â explanation is going to be hard to follow. This is really difficult to understand.

Â This is extremely, extremely hard. And, and the goal of this course actually

Â isn't for you to understand this, the goal of this course is to give you the

Â machinery so that you're now able to take a whole semester course on real analysis

Â that you're enduring, which you should be able to understand that.

Â So, I wouldn't worry too much if you can't understand this.

Â The point is if you could understand the first part and understand how to deal with

Â the formalism you now have the machinery you need to understand this is very deep

Â conceptual definition from within mathematics.

Â Okay, and so to have reached such stage in 4 weeks I think is pretty, is pretty

Â remarkable. The hard part is to come.

Â And for those of you that want to go on and study more mathematics sooner or later

Â and hopefully sooner you're going to need to master this definition and what it

Â means. Okey dokey?

Â Well as far as we're concerned we're done with this one.

Â