 Welcome back to our lecture series Math 3120, Transition to Advanced Mathematics for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. This video is the first video on lecture nine, and in this video, we want to talk about the idea of an index set. To describe an index set as, we have to first remind ourselves on this sigma notation that we probably have seen previously in maybe a place like calculus. We often see this notation where you see this capital sigma, there's some stuff on the bottom, there's some stuff on the top, and then there's a sequence of numbers right here. This Xi is going to represent the general formula of some sequence of numbers, and it's going to depend on some variable, which in this case, we've indicated that as i. That i, you also see here on the bottom, and represents the index of the sum. It's just a dummy variable that keeps track of which term in the sum are we in. You're going to see a number right here, which then tells you the initial term that is the first term of the sum, and then you see a number on the top, that's then going to tell you the terminal term, that is where does the sequence end. Now, these can be numbers, they can be variables like you see in here, it could be infinity for example. This in particular tells you where you start and where you stop, and then what is starting and stopping the number i, which you find in the formula right here. This sigma notation then represents an abbreviation of the sum, x1 plus x2 plus x3 plus x4 plus x5 all the way up to xn. And so this sigma symbol right here is then used to indicate you're taking this generalized sum. Now the Greek letter sigma here, you can think of as like the Greek alphabet, it's the third equivalent of a capital S. And a capital S here stands for sum. S is for sum here, and so sigma is just standing for a sum. It's a pneumonic device that when you see this sigma it's indicating to you're gonna add together some terms. Which terms? You're gonna add together these terms as they range from one to n like here. You see this all the time in calculus like when you talk about Riemann sums or power series, infinite series, things like that. Something that shows up less often in calculus is the counterpart of this notation when it comes to products. So if you see something like this, the notation looks exactly the same. You have a sequence of numbers, which we're calling xi. You have some index listed, we're also calling that i. And then this number tells you where you start the sequence. This number tells you where you end the sequence. And these boundaries could be themselves, specific numbers, variables, they could be infinite. The only thing that's different between this notation instead of a sigma, you now have a capital pi. Now we often think of the number 3.14, et cetera, when you think of pi, but this is a capital pi. And pi, the Greek letter pi in their alphabet is actually equivalent to the Roman letter P. It makes that puh puh sound. And so I want you to think of this capital pi as likewise a mnemonic device. Pi stands for product. And so unlike the sum, we're gonna combine together the end numbers because we have x1, x2, x3, x4 up to xn. With the sigma they're combined together using addition because it stood for sum. Now the product means we're gonna combine them together by multiplication, x1, x2, x3, x4, all the way up to xn. And it turns out that we're actually quite used to this idea of putting together products. We see them all the time, like for example, even if we haven't seen this pi notation before, we have probably seen things like in factorial. Now what is in factorial? In factorial is then the product of the number sequence i as i ranges from one to n. If you expand this product notation here, you would get one times two times three times four all the way up to times by n. And that's exactly what in factorial is. When you think of an exponent, for example, take for example, the number three to the fourth power. Or we'll even do just three to the n. This you could think of as the product of the constant number three as i ranges from one to n. This type of repeated products show up all the time just like we've seen repeated sums. Again, you just often don't see the notation. So I want to introduce that in this video. So we're used to that, oh, if you want to generalize an operation, you just use this big symbol, a big symbol to represent it. The numbers on the bottom tell you where you start. The numbers on top tell you where you end. This is the thing we're operating on. And here is the index of that situation. Now I should also mention that in order to make sense out of these things, we do need that our operations be associative. Associative here meaning that how you do parentheses does not matter. Notice that if you take x one plus x two plus x three, this is the same thing as x one plus x two plus x three. Addition and multiplication are binary operations. They're defined to take in two input and you make a sum or product from those two numbers. If you have an associative property which both addition and multiplication do, you can actually drop the parentheses and you can make it be any combination. You can add three, four, five, 12 terms together. It doesn't matter. There's no ambiguity whatsoever when you do that type of calculation. Now our goal in this video, in addition to reminding us about the previous notation, is actually to extend it to the setting of set theoretic operations. In particular, there's two operations we've played around with a lot in this lecture series already, the notion of intersections and unions. And so we can actually construct a generalized intersection and a generalized union. What does it mean to intersect three sets together, four sets together, five sets together, insets together? This generalization, because intersection is also associative, we don't need all of the parentheses, we can drop them down. And so to be in this intersection, a one, a two, a three, up to a n, what you're looking for is an element that belongs, it belongs to all, excuse me, it belongs to all of the sets in the collection there. And with regard to unions, how do you take a generalized union? How do you take the union of a one, a two, a three, a four? Well, again, we don't need the parentheses because the operation is associative here. If you take the generalized union here, you're looking for all those elements that belong, it belongs to at least one set in the collection. And these words, all and at least are things we're gonna talk about in the future. This is related to the notion of a quantifier that we're gonna talk about in the next lecture, not in this video, of course. So when you have a, you can take a large intersection of sets or a large union of sets because of the associative property, but it can get a little bit tiresome to write all the sets together. So what we can do is we can abbreviate the large union, mimicking the sigma notation we have before. So we're gonna take the union of sets that belong to our sequence, the AIs. And then the index represents the general term, the general token of these sets here. It'll range from one to N. Now, unlike addition where we use sigma as a mnemonic device and with multiplication, we use pi as a mnemonic device, there's so many operations under the sun that we can't come up with mnemonic devices for all of them. So oftentimes what you do is you just take the symbol for the operation and then you just gigantimax it. You just make it a bigger looking symbol. So you have a little union and then a big union. And so this means that you're gonna iterate the union over the range one to N. And likewise for intersections, you have a small intersection and you have then a big intersection. The big intersection means that you're gonna take an intersection of a collection of sets, I mean, where you range from one to N, like so. Let's do an example of such a thing. Here is our family of sets. We're just gonna do something simple while three sets right here. A1 is the set that contains the natural numbers zero, two and five. A2 contains the numbers one, one, two and five. And A3 contains the numbers two, five and seven. And so then if we wanna compute the intersection of the AIs as I ranges from one to three, this is just the abbreviation of the form A1, intersect A2, intersect A3. So we're looking for those numbers that belong to all of the sets in the family. Now, when you look at A1, zero is contained in A1, but not in any of the others. When you look at two, two belongs to A1, it belongs to A2, it belongs to A3, it belongs to all of them, so it makes it into the intersection. If you look at A5, excuse me, five is in A1, it's also in A2, it's also in A3, so sorry, that was a seven, there's the five. So five belongs there as well. So the intersection will contain two and five. But if you look at the other elements, one does not belong to every set, seven does not belong to every set. So the intersection will just be two and five, that pair right there. What about the union? Well, if you want to look at this right here, this is shorthand for the union of the sets AI as I ranges from one to three, in less compact form, this would look like A1, union A2, union A3. So we're looking for all the elements that belong to at least one of the sets. So you can go through each of the sets one by one. A1 has a zero, so does the union. It has a two, so does the union. It has a five, so does the union. Move on to A2, A2 has a one, so will the union. Two and five we've already considered. Then you come over to A3, two and five we've already considered, but seven we haven't, so we join it to the union. And so the union of these three sets will then be the five elements, one, zero, one, two, five, and seven. Those are all the elements that belong to at least one of the sets. Now it's also important to note that with the sigma notation we saw on calculus, it was often the case that we have like an infinity on top. We have an infinite sum, like with Riemann sums, the idea to find the area into the curve is to take an infinite sum where your rectangles get skinny, skinny, skinny, or with power series and the chlorine series and the like. When you have a series, you have an infinite sum. In that situation, the infinite sum is defined to be the limit of the partial sum. So we get something like sigma as i ranges from one to infinity and you have like an xi right here. By definition, this is the limit as n goes to infinity of the partial sum i equals one to n of the xi is like so. So that's how we define an infinite sum as the limit of partial sums. We can do the same thing for intersections and unions as well. So if you'd see something like this right here, we're gonna take the infinite intersection as i ranges from one to infinity of the collection of sets ai. This is then defined to be the limit as n goes to infinity of the partial intersection where we range only from one to n. And as n is allowed to get arbitrarily large, then we can take the limit, thus giving us this infinite intersection over here. We do the same thing for the union symbol as well. If you see an infinity on top, you should then read it as i take the union as i ranges from one to infinity of the ai's. This then means we take the limit as n goes to infinity of the partial unions. We're now i ranges from one to n. And thus we then let n get larger and larger and larger going towards infinity. And then the union is gonna be the set that's then derived by letting n get bigger and bigger and bigger and bigger. So let's look at an example of this as well. Let's look at this time at this family of sets of the following form. A1 is gonna look like negative 101. A2 is gonna look like negative 202. A3 is gonna look like negative 303. And then if i is any positive integer, just for by way of notation here, we've seen the symbol like this before. It's the integers. The superscript plus just means we want the positive integers. So one, two, three, four, five, six, et cetera. So for any positive integer, we can define the set ai to be negative i, zero i. So for example, a5 is just the set negative five, zero and five. That's our formula right there. And so then we have this. Notice we now have an infinite family of sets because these sets are indexed by a positive integer. And so we then could ask ourselves, what is the intersection of this family as i ranges from one to infinity right here? Well, the ai's are just these sets right here, negative one, zero, it's not, sorry, not negative one, negative i, zero i. And so the intersection would then be what are the elements that belong to every single set of this form? Now when you look at just the first three, a1, a2, a3, the only thing that belongs to all of them is zero, right? Cause a1 has one and negative one, but a2 doesn't. a2 has negative two and two, but a3 doesn't. a3 has three and negative three, but a2 and a1 don't. They all have a zero. And in fact, when we look at the general formula, they all have a zero as well. And so when you take the intersection of this infinite family, the only thing that will belong to all of the infinitely many sets will be the number zero. So the intersection is just this singleton. Conversely, they'll look at the union here. If we want to take the union as i ranges from one to infinity of the family of sets, a i, this is the same family of sets, negative one, zero, sorry, negative i, zero and i there. We then want to take the union. So if you belong to at least one of the sets, you belong to the union. When you look at the a1, you're gonna get zero, negative one, one. When you then unite with two, well, you're gonna still have zero, that was already there. You got two and negative two. When you unite with a three, again, zero is already there, but now you gain three and negative three. So then if you were to unite with a four, you're gonna gain plus or minus four. When you unite with a five, you're gonna get plus or minus five. When you unite with a six, you're gonna get plus or minus six. And in general, when you unite with a i, you're gonna gain the elements plus or minus i. So you get both the positive and the negative integer there. And so then we can see that this union is equal to the set of integers. Every integer is represented somewhere. If you take some integer z, then n is going to then belong to the set A, absolute value of n, because n could be a negative number, right? It'll be inside of there. You get this negative n, zero, n in that situation. I guess the only potential issue would be if n is zero, but zero belongs to each and every one of these sets. So that's fine too. So the union of all of these sets does turn out to be the integers. This was an infinite union and an infinite intersection. Now we've talked a lot about unions and intersections. I should mention also that this notation applies for Cartesian products as well. Now a Cartesian product, we think of actually as multiplication of the sets. And so the symbol we use, instead of like a giant x, you actually use the product symbol that we introduced before, the capital pi. So when you look at this notation right here, this means we're gonna take the product of the sets, A i as i ranges from one to n in that situation. Well, if you're taking a product of sets, this is gonna be the Cartesian product. So this by definition is then the Cartesian product, A 1 cross A 2 cross A 3 cross A 4, all the way up to A n. Now we've talked a lot about the Cartesian product of two sets. This is by definition the set of all ordered pairs where your first coordinate comes from the set A and your second coordinate comes from the set B. So we write that as something like the following. But we can generalize the notion of a Cartesian product to then if you take the, if you take a Cartesian product with n mini sets, you then take the set of all list of length n. And remember as we introduced list in the beforehand, a list by definition is ordered. So you don't have to call it an ordered list, all lists are ordered. And so the first element of the list is taken from the first set. The second element of the list is then taken from the second set in the product there. All the way down to the nth term of the list comes from the nth set right there. That's what we describe generally right here. AI comes from capital AI. And so this is the set of all possible in length list where elements in the list are taken from the respective sets right there. Thus generalizing this ordered pair of bits as we had seen before. Now this product, we can have arbitrary Cartesian products but one that's extremely important is when you take a set cross itself, cross itself, cross itself and you do this like say n times. Now like we talked about before, Cartesian product we're thinking as a multiplication of sets. And so what you can then do is you can define that if you take the in fold Cartesian product of a set with itself, we then call this the Cartesian exponent. And we use exponential notation here. A to the n means the Cartesian product of a n times. And so then what does a typical element here look like? I guess I should write this the other way around. Nope, I was right the first time. A typical element here would be a one, a two, all the way up to a n. So the set a n is then just in length list of elements coming from a. And this is something we actually see a lot like in geometry, maybe calculus or algebra if you're doing some analytic geometry. In linear algebra, you see these types of notation used all the time. So for example, the set R2 is by definition the Cartesian product of R with R. And so this would then be the real plane where the elements of the real plane look like ordered pairs X, Y. Those are the coordinates we give. Like in linear algebra or multivariable calculus setting, you might also have seen R3. This by definition is R cross R cross R. And this gives us three space where elements in three space look like ordered triplets. An ordered triplet is just a list of length three, of course. And so this is notation we've seen before and we will be expecting to see things like this, of course, in the future. Now, before we end this video on index sets, I do want to then describe why do we call them index sets? It turns out all of the sets we've been considering with these unions and products and intersections, your index always turns out to be a natural number. It turns out that is not required. There are times where it is inconvenient to index a family of sets using natural numbers. And there's also times where it's impossible to use the natural numbers to index a family of sets. And so the titular topic of this video here, the index set is exactly that. If you take a set I, we call it the index set. If the elements of that set are used as tokens, markers for sets in some family, okay? So consider you have this family of sets, A sub I. Well, that I we put as a subscript there, it's just a marker, right? It just tells you like this is the identification number for that set. And then the index set I is just then the collection of all of the identification numbers that are possible. And this set could be a set of natural numbers. It could be something much, much, much, much bigger than the natural numbers though. And so if you then talk about the intersection of a family of sets indexed by the set I, we write things a little bit different. Instead of having something on the bottom and something on the top, we actually just put it all on the bottom right now. And so then when your top is blank, you should be thinking of, oh, the index set is on the bottom. For which this right here is still our index. This is a generic arbitrary marker. This right here though is the index set. This is the set of all possible markers you could use to describe the sets right here. You still see a big intersection symbol, which means we're gonna intersect something. What are we gonna intersect? This family of sets right here, where again that I is used to indicate the typical element, the typical marker of this set. And so by definition, this intersection is then all of the elements that belong to all of the sets here. And this is the keyword here, all. The intersection grabs those elements that belong to all of the AIs. Now, conversely, if you use a big union, you then see the index set here on the bottom. So this means we're gonna take the union of sets of the form AI, where AI is indexed by the set capital I itself. And this by definition is gonna be all of the elements X, which belong to at least one of the AIs. And that's this keyword here as well, at least. At least one describes unions, all describes intersections. And we will see this in the next lecture, this idea about quantifiers. For all is one of these quantifiers, the so-called universal quantifier. And then the other one is the existential quantifier for at least one. We'll make some more sense out of those next time. Now, what I wanna then do is then consider an example of a infinite union and intersection where our index set is something other than the natural numbers. Because when we looked at those intersections and unions before with the infinity on the top, you took like I equals zero to infinity AI. I want to make mention that using the notation we just introduced, this should just be like, oh, I belongs to the set of natural numbers of the AIs. So ranging from zero to infinity just means you're just ranging over, your index set is just the natural numbers. Or if you range from one to infinity, you're just ranging over the positive integers. So those are just special cases of what we're considering now. Now, in this example right here, what we're gonna do is we're gonna take our index set to be the interval of real numbers between zero and two, inclusive, so zero and two are both inside that. This is not an interval, there's not a set of, it is an interval. It's not a set of natural numbers. It only contains three natural numbers, zero, one and two. There are a lot of real numbers in there, both irrational and rational. Now, using this index set, we're gonna describe a family of sets. We're gonna call them AI, and AI is gonna be abbreviation for rectangles. Remember, a rectangle is just an interval, cross an interval, and we can view this as a subset of R2. Now, the first interval is gonna be the interval I to two, where I and two are the X coordinates here. I, of course, is the index of the set. And then the second interval will be zero to I, where this is now describing the Y coordinates. Again, I is the index of our set. So this is gonna be a rectangle whose vertices are I zero, two zero, two I, and I I, which I have an example of one illustrated here to the right. So if this is the rectangle associated to index I, it'll contain as a vertex I zero, two zero, two I, and I I that you see illustrated here on the screen. And now as I ranges between zero and two, you can get lots of different rectangles. Here's such a rectangle, another one would look something like this. And so you get these different rectangles based upon your index I. So then what I wanna consider is what would be the intersection as I ranges on the set zero to two of all of these AIs. All right, can we compute this thing? Now, I wanna make mention that this intersection is going to be a subset of any individual set in there because if you belong to the intersection, you belong to all of the sets. In particular, you belong to a particular set. And so if you look at A zero, pause the video to compute this if you have to, but convince yourself A zero is just the interval zero to two along the x-axis. And likewise, I wanna mention that, let me write it this way. If you consider the set A one, this would be the interval, I'm sorry, A two. This would be the interval because A one would be this rectangle right here. A two would actually be this line segment that's vertical right there. If you look at these ones, what do these two sets have in common? The only thing they have is this point's two comma zero. Now, two comma zero belongs to both of these sets. If it belongs to the intersection, it would have to belong to each and every one of those. And you can convince yourself that two zero does in fact belong to each and every one of these rectangles. And because of the two sets we just consider, that's the only thing that belongs to everything. So the intersection will just be the single 10, two zero in the plane. Now, let's go the other direction. Let's take the union now of these sets as I ranges from zero to two in the interval. Take the union of all of the AIs. So what we're gonna do is we're gonna take every point in this rectangle. We're gonna take every point in this rectangle. We're gonna take every point in say this rectangle right here and we allow these rectangles to vary. Now you might notice on the screen here, I have this triangle drawn like so. Whoops, try that again. Call this triangle T. I claim that the union of all of the AIs is actually equal to the triangle T itself. Now clearly by construction, each of these AIs is gonna sit inside of the triangle. So that gives us inequality in one direction. But for the other direction, I want you to note here that if I take any element inside of the triangle, call its coordinates X comma Y, I can then draw a rectangle that contains that point. In particular, I can take a rectangle where that point is actually on the line segment on the top of it. And so every point of the triangle belongs to at least one of the rectangles and therefore belongs to the union. This then shows us that the union of all of these rectangles is in fact equal to a triangle. And so this is a really fun example to end on here that it shows us how infinite unions and infinite intersections can be indexed by anything. Doesn't have to be natural numbers and how one can actually calculate these things. Notice here when we talked about this, I made an argument that containment went in one direction and then a different argument that containment went in the other direction. If two sets are subsets of each other, that actually makes them equal to each other. This is a proof template we saw beforehand. But isn't this a calculation? Didn't I calculate this set? I wanna point out here that it turns out proofs and calculations are really the same thing in mathematics because all they are is just mathematically correct communication that expresses mathematical truth.