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this video we're going to talk about infinity there are different kinds of infinity there's countably infinite and there is uncountably infinite and we're going to talk about both these kinds of infinity here before we get started we need to remember some terminology about sets first of all if a set is finite then it has a finite size and we can talk about how many elements are in the set a set that is a a finite might have for example 17 members and we can just count the number of elements in the set but when we start talking about sets that have infinite sizes with an infinite number of members we need to be a little bit more careful so I want to remind you about this terminology if you haven't seen it before or go over it if you have seen it let's assume that we've got a function f and it maps elements from one set the set a to some other set the set B we call set a the domain and B we call the range so here you see a picture and this is set a with some elements in set a and this is that be okay and f is taking each element in a and going to some element in set B now sets going to act different ways on their domain and range and we have these terminology these terms one-to-one and onto and correspondence let me go through each one of those in turn first of all if a function is one-to-one what that means is that every element in a is mapped by the function to a different element in B so here you see some elements in a two elements in a and they're getting mapped onto the same element by F and that's not one-to-one okay here I'm saying it and I'm using a and B which is slightly confusing because here I'm also using the letters A and B perhaps I should say and why here instead of calling these things little a and little B but if two elements are different then when you apply the function to those two elements you'll get a different element in B if two elements are different like these two elements then when we apply the function to them we'll get two different elements elements that are not equal so we do not allow something like this we don't allow arrows to touch each other other heads now that's one two one on two is something similar in that case we look at all the elements in the range remember a is the domain B is the range of F and we look at all the elements in the range in set B and we make sure that every one of them can be reached from an element in a so we could say that this that this way every element in B is hit it there is an element in the domain that will correspond to the element in B when applied to when the function is applied to that element we'll be able to get to that element and B so we do not allow situations like shown in this picture and finally we can define a correspondence and this is a picture of a correspondence any function that is one-to-one in other words these lines don't intersect and onto meaning every element of B is touched then we have a correspondence for element every element in a there is a corresponding element in B and for every element in B there is a corresponding element in a so this function if it is one-to-one and onto it creates a correspondence between the two sets now we can start to talk about sets whose sizes are infinite if the set has a finite size then we can simply count its size and if we can count the size of a set then we know it is a finite but if the set has an infinite number of elements we begin to have difficulties because we can no longer count the number of elements in the set Georg Cantor came up with this very useful idea which is to say that two sets are said to have the same size if and only if there exists a correspondence between them in other words if you can find a function that is a correspondence between the two sets then we say that those two sets have the same size so this is really a definition of what it means for two sets that are infinite in number to have the same size and our second definition that's useful is we say that we say a set is countable if it has either a finite size for example if it has 17 elements its countable or if it's infinite and there's a correspondence with the natural numbers remember the natural numbers are start at 1 and go on up 1 2 3 and so on so we talked about accountably infinite set if we can put it into correspondence with the natural numbers so here's an the first example of this is the set of odd numbers okay 1 3 5 7 and so on this set is countably infinite because we can put it into a correspondence with the set 1 2 3 and so on in fact let me show that correspondence ok here are the odd numbers 1 3 5 7 9 and so on and here are the natural numbers which I made with a little fancy in there 1 2 3 4 5 and you can see there's a correspondence between them okay you can figure out that function it's a relatively simple arithmetic function and so there is a correspondence between the natural numbers and the odd numbers but also notice with the odd numbers for a subset of the even numbers 2 4 and so on are not elements of the set of odd numbers but they are set they are in the set of natural numbers so we can have a situation where we have a set that is both has both the same size as the natural numbers and is at the same time a proper subset of the natural numbers now the next nets that we want to look at is the set of rational numbers a rational number is a fractional number that would include decimal numbers that don't have repeating digits at the end for example 0.3 which is early 0.3 zero zero zero zero or one third which is 0.3 3 3 3 3 3 3 etc those digits repeat those are rational numbers and every rational number can be expressed as a fraction 0.3 is really 3 over 10 point 3 3 3 out to infinity is really 1 over 3 so a rational number can be expressed as a fraction M over N where m and n are both natural numbers I suppose we should properly also include the negative rational numbers which this doesn't include but we can kind of ignore that for now this set is countably infinite and we're going to show on the next slide a proof that it's countably infinite okay we're going to show a correspondence and then the next thing we want to look at is the set of irrational numbers ok this set includes numbers like pi and other decimal note and other numbers whose decimal expansion contains an infinite sequence of digits that does not repeat so pi the square root of 2 and so on are examples of irrational numbers and there are also many many more in fact we'll find out that they this cannot be could it put into a correspondence with the set of natural numbers so this set is uncountably infinite i just said that the set of rational numbers is countably infinite and in order to prove that we need to find a way to put them into a correspondence with a set of natural numbers in other words we have to list them we have to be able to create a list of every single rational number in such a way that every rational number is included ok here i'm repeating the definition of the set of rational numbers and i have left out the negative numbers which properly speaking native there are negative rationals as well as positive rationals but we're just looking at the positives here and so i need to come up with in order to make this proof complete a correspondence here I've shown the natural numbers 1 2 3 4 and I've shown a few rational numbers ok and so I'm suggesting the idea of a correspondence and there's no system to what I've done here and this is not actually showing a correspondence it's just showing the kind of thing that I need to find in order to show that the set of rationals is countably infinite but down here I'm going to show a systematic way of listing every single rational number I'm going to show a procedure for listing rational numbers and that procedure will eventually list every rational number and the order in which I list them will show the correspondence between the rational number and a natural number so what have I done I have made it a table and this table is infinite in both directions so it it's 1 2 3 4 5 6 7 dot dot dot and down here dot dot dot and at every point in the table I've taken the row number and divided it by the column number so Row two column three is 2/3 or 2/3 so if you give me a rational number any rational number that can be expressed as M over N and all rational numbers can be expressed as a fraction M over N then it will be in this table okay for example if you ask about seventeen four hundred and fiftieth well that's going to be on the seventeenth row the four hundred and fiftieth column okay it will be in this table so this table is infinite in size but as I start to draw this table I will eventually hit every single rational number okay if I want to draw this table I'm going to start in the upper left hand corner and I'm going to go down and to the right at the same time sort of the way I've done here I'll never reach the right edge I'll never reach the bottom edge because it's infinite in both directions but I can just keep growing this table and now that I know how to create the table or at least a part of the table I can start enumerate in the rational numbers so what I'm going to do is I'm going to go through this table on the diagonals like I've shown here in this red line okay and as I do that I'm going to hit every element in the table that's an infinite process but I'll hit every element okay if you pick some rational Oh more like seventeen four hundred and fiftieth eventually I will hit that number I will hit every rational number eventually and I'm going to list them out okay the first rational number is one over one that's just equal to the number one so I print that out and then I print out one-half and then I print out two I can simplify if I want to and then I keep going and I print out three and then here 2 over 2 is the same as 1 and I've already printed one so I can skip that one if I want to and then keep going 1/3 I haven't printed out then I print out 1/4 2/3 three halves and 4 and then 5 over 1/4 over 2 is really the same as 2 I already printed that out 3 over 3 is the same as 1 2 over 4 I've already printed that one out 1/5 I have not printed out and so I print these numbers out I print out the rational numbers in this order this this this this this this and so on I keep going and eventually I will print out every rational number Oh any rational number you give me I will eventually print it this forms a correspondence and this shows that the set of rational numbers is countably infinite ok now let's look at the set of irrational numbers it turns out the set of irrational numbers is uncountable infinite there are more many more irrational numbers than there are natural numbers so what are irrational numbers well we can represent these numbers with decimal expansions and here I've shown a few examples are familiar pi 3.14159 really these things have infinite decimal expansions and they never repeat so I should add the dots here and I've shown square root of 2 every whole number has either an integer root in its square root is either a whole number or it's an irrational number with an infinite decimal expansion that never repeats likewise the constant e boiler is constant is two point seven one eight two and so on and that's an infinite expansion and there are many more and here's one I just made up okay this is no number that anyone's ever thought about before it's five point six seven nine and it's some number you know with an infinite number of digits and those digits never show a repeating pattern okay now here are some rational numbers it to look at in contrast one third is expressed in decimal is point three three three three three and oftentimes we draw a line over the lot over the last digit to say that it repeats so in other words it's 0.3 with an infinite number of threes okay it can be expressed as a fraction or as none of these numbers can be expressed as a fraction of whole numbers now between any two rational numbers there are an int there's an infinity of irrational numbers and I've tried to show that by showing a rational number one third and here's another rational number that's almost the same it's very very close but at some point the digits differ okay in this case it's it I've drawn commas here to make it easier to parse but in the sixth position okay this thing differs by one from one third and then since it's rational it has to have repeating and I try to keep it as small as possible after making after going up one digit here so I use just zeros out here so this number is a fraction it's 333333 close to a third but it's not exactly a third it differs in the sixth position and it has like any rational number it repeats but between these two I can squeeze in an infinite number of numbers okay in this case I am showing a number that is larger than 0.33 3 3 3 3 okay it's larger because in the seventh position it's five but it's smaller than point three three three three three four because in six position it's a three so and you can end it just I don't list the digits out here but you can see that between these two numbers there is a number that will fit in there and I suggested that there's an irrational number of course there are rational numbers as well but this is to show that between any two rational numbers there's an infinity of numbers and in particular there's an uncountably infinite number of irrational x' so now how can we prove that the set of irrational x' is uncountably infinite well the answer or the approach to the proof is to prove it by contradiction assume that there are accountably infinitely many irrational numbers so if they're countably many numbers that are irrational then it stands to reason that we can put them in a correspondence to the whole numbers so here I have drawn a correspondence I'm using first of all I want to just look at the numbers between 0 & 1 so I've selected some irrational numbers between 0 & 1 the first one is pi divided by 10 right 3 1 4 1 5 9 the next one is the square root of 2 divided by 10 and then Y divided by 10 and then the number I made up divided by 10 and then some other numbers okay but the idea is that any irrational number between 0 & 1 can be listed out and if there are accountably many of them we can list them out in some order such that there's a 1st 2nd 3rd 4th there is some sort of correspondence so we're assuming there are accountably many irrational numbers so in if that assumption is true which it is not but if it were true then we could make this table like this and and we could list these things in order and we could put them in a course by Condon's with the natural numbers now then the trick here to show to the contradiction is to look at the diagonals and so this is called a proof by diagonalization technique if we look at the diagonals in this table taking the first digit from the first number the second digit from the second number and so on we can construct a new number okay so what I've done here is I've added 1 to the 3 to get 4 I've added 1 to the 4 I've added incremented by 1 in the case of 9 I subtract 1 the case of 3 I've added 1 and I've been adding 1 the point here is I'm making a number that differs each of these digits differs from the corresponding digit in the diagonal so this number is clearly not equal to the first number because it differs the way I've constructed it it differs in the first digit and it's not equal to the number in this the second number 0.14 142 because it differs in the second digit and it's not equal to the third number because it differs in the third digit and so on and in fact the number I am constructing this way which of course has an infinite number of digits has a difference between it and every other number in the table so this is a number that can't possibly be in this table yet this is a rational number sorry yet this is a perfectly valid irrational number so we've got our contribution since this number is an irrational number but it's not in this list then we've got a contradiction because we said we were listing all the irrational numbers so that is the proof that the set of irrational numbers is uncountable infinite