Section 2.1 Part 1

hello everyone in this video I’d like to introduce a new topic set theory we’ll begin with some historical background and the context for why we study sets then we’ll talk about what sets are and how we mathematically describe them I’d encourage you to have your math and our world textbook on hand and pause this video to try the examples on your own as needed I also wanted to point out that the textbook offers a try this one problem after each example in case you need even more examples than those in this video we’ll begin with section 2.1 from math and our world who developed set theory unlike other subjects in math we can trace the development of set theory back to primarily one person Georg Cantor he was a mathematician who tried to answer some questions about infinity and one of the things that he’s known for is is actually proving that there are more real numbers than rational numbers rational numbers are fractions and even though we have an infinite number of both of those types of numbers he showed that there’s actually an infinity that’s bigger than another infinity you can imagine that this work was groundbreaking at the time it was not well-received and tragically his the criticism by his peers eventually led to brighten ervice breakdowns and he ended his life in a mental institution thankfully we now widely accept his work and it’s it’s too late for him to get the recognition that he deserves but we can attribute all of our progress in set theory back to can’t or if you’re interested here a couple books on the topic the one on the left is a biography of Georg Cantor and it also contains quite a bit of information about infinity questions dealing with infinity definitely the philosophical theories behind his work and just different questions about infinite sets as well the book on the right is more of a general mass kind of a book very readable also and this one has short answers more like a paragraph for just some some questions that you might have on any really any math topic I’ve pulled a few excerpts from this one to give you an idea of the background for why we study sets set theory is one of the foundations of mathematics so what it what it does is it allows us to have a way of organizing or talking about math and then we could use that to go on and study other types of math it’s also a philosophical topic and is of interest to philosophers and as you can see here this excerpt says that the foundations of math provide ways for philosophers and mathematicians to try out general philosophical doctrines in a specific scientific context so if you ever thought that math was black and white and that you know we agreed on you know all the fundamental fact since it and things well that’s not entirely the case and there is you know there are some different approaches based on someone’s philosophy well what is set theory set theory is kind of redundant here the mathematical theory of sets but basically one goal or one aim of set theory fits in with the aims of logic research to find a single formula theory that will unify and become the basis for all of mathematics so just as with many other scientific theories even something like string theory you know people are constantly searching for a way to to unify and to to find a way to describe the universe and that’s one of the aims for set theory as well okay so let’s start in on on section 2.1 then this is the beginning of the information from math and our world I’m some basic definitions a set is just a collection of objects asset is well defined if for any given object we can objectively decide whether it is in the set or not in the set so for example the set of letters in the English alphabet that is a well-defined set because there are 26 letters and if we’re given a letter we can you know know whether or not it is in the set for sure it one way or the other a set of great musicians this is a set but we wouldn’t really consider it to be a mathematical set and it’s definitely not well-defined this one is subjective you’ll have many different opinions on who should or should not be

in that set an object in the set is called either an element or a member of the set so G is an element of the set of letters in the English alphabet but the Greek letter beta is not a member of the set of letters in the English alphabet ok so element member words for things in the set there are three ways to describe sets we can list out the elements in the set we can give a verbal description or we could use a more technical set builder notation and you’ll see this one in formal math writing more often it uses more symbols it’s a shorter way to write very long descriptions another way to name a set is to use a capital letter so you’ll see things labeled like said a or said M the roster method or the listing method those are just two words for the same method you list the elements between braces those are the curly kind of these things right with commas between the elements elements can be listed in any order and repeat elements are only listed once so for example we have set be here ok and I can tell that that’s the name of the set because it’s a capital letter and we write set B equals and then in curly braces you just kind of give a roster or a list of all the things in that set txw one and five so there are five elements in that set and then notice that we can list those in different orders but it still represents the exact same set all right so it’s not like for example an ordered pair and in algebra when you’re graphing like the point two three that’s a different point than the point three to pay but incest doesn’t matter it doesn’t have to be numbers all right so for example set ass here said s is the set of that square symbol a heart symbol and a star symbol all right repeat elements are only listed once so if we had had maybe this is the set of objects on the page of a coloring book and maybe there were two stars well when we go to write the said you only write the star once all right let’s practice and pause if you want to try these out on your own and then if you want you can use this video as a way to to check and see if your answers are correct you could also just read along in the textbook right the set of months of the year that begin with the letter M okay so let’s do this using the the roster or listing method so you curly braces and then you just list out the things in this set so months of the year that begin with the letter M what we have March and we have may all right so you put a comma between those two all right and then a closing brace at the end right the set of letters in the name Jennifer right braces all right and capital versus lowercase doesn’t really doesn’t really matter it’s still a J there may be situations where you would be picky about that but I’ll take either one okay so we have a J we have an E we have n now there are two ends but we don’t write it twice you just write it once hey I f a second e we already have an E and end our and that’s it to specific sets or I’m sorry that your textbook will start to just assume that you know the set of counting numbers or natural numbers two words for the same set these are the you know kind of naturally these are the numbers that you would probably learn first if you’re a little kid and learning to count so you usually start with 1 1 2 3 4 5 6 7 e etc and all of those numbers are the set of natural numbers and we use this little scripted n to represent the natural numbers so it’s an end but it has like an extra oops an extra little bar to it get it one of these times okay and then the sets you know other text books may not use these letters for these sets but in this textbook when you see an e a capital e that stands for the set of even natural numbers 2 4 6 8 10 12 14 etc and then o stands for the odd once 1 3 5 7 9 etc okay notice that 0 is not in

this set so often when kids are taught to count they don’t start with 0 and natural numbers don’t either all right let’s practice use the roster method to write the set of natural numbers less than six okay so less than means not actually including so less than six we would start with five and then kind of go down from there and we need to know where to end at the other end well the first natural number that the smallest one is a one so natural numbers less than six would be one two three four and five hey the set of odd natural numbers greater than four okay well the first odd one greater than 4 is 5 5 7 9 11 and then we use an ellipsis just to indicate that this keeps on going and then close the the brace at the end the set of natural numbers between 6 and 13 okay so between does not mean including so if you put a 6 and a 13 in this set that’s incorrect between 6 and 13 would be 7 8 9 10 11 and 12 and the set of even natural numbers from 80 to 90 now this one does mean starting an 80 and ending at 90 okay so from include those two end ones all right so starting at 80 and then we need even ones all the way up to 90 88 and 92 new symbols the symbol this little kind of a curve here means an element I sought to use to show that an object is a member or an element of a set all right so if you’re drawing this I’ll write it a little bigger it’s very curved with a line in the middle okay example let a be the set of days of the week then monday is an element of set a ok so the thing on the left is the element thing on the right is the set the symbol kind of that e with a slash through it is used to show that an object is not an element of a set ok see you write your Isom bowl slash through it means not an element so march is not an element of said a because that a was the set of days of the week all right true or false statements Aragon is an element of set a where a is the set of states west of the Mississippi River ok true ok Part B all right well 27 isn’t shown there in that list but there’s an ellipsis which indicates that the set continues so we need to check a few more terms and see if eventually 27 would be in there alright so we need to find a first and it looks like these numbers we have a pattern of adding 4 to get from each one to the next one so after 17 would be 21 and then 25 and then 29 and it would keep going but we can see from that that we’ve skipped right over 27 so 27 is not in that set okay z is not an element of the set containing v w x y and z that’s false Z’s in the set okay Part D zebra is an element of the set containing Zeb are a-okay kind of a trick here but zebra for looking at this as an element that means the whole word as you know as an entire word all right any we don’t actually see the whole word in the set we just see single letters making up zebra so this is also false three is not an element of 12 13 14 15 that is true so even though you see a three in 13 3 is not an element of the set there are four elements they are the numbers 12 13 14 and 15 okay so we’re in the middle of talking about three ways to describe sets we’ve talked about the listing method a second one is the descriptive method and this one uses a short statement to describe the set the description should be clear enough that someone can reconstruct the exact same set from the description alone so if there’s a possibility that your description leads somebody to write a

different set than the line you intended then it’s not a correct verbal description okay which of the descriptions below accurately describe the set containing two three and four alright so kind of take a minute to look through these and see if any of those seem like they would be good enough okay the first one the set of numbers between one and five all right well our numbers are between one and five but this description could also maybe lead somebody to write how about 23.5 4.3 well that’s also a set of numbers between one and five all right so we have to have something in our description that that indicates what type of number it is because just the set of numbers between one and five would include all the decimals and fractions and you know irrational numbers like the square root of two and things like that Part B the pattern is counting up by ones okay well it’s a true statement but it doesn’t describe this set because first of all it’s not kind of like a noun phrase the set of something it’s just describing the pattern not the set and then second of all it also doesn’t tell us where to start an end all right the pattern is counting up by ones well if i write this set 10 11 12 13 that pattern is also counting up by once all right so that one’s not accurate enough the pattern is counting up by one’s from two to four that is a true statement but we wouldn’t say that this is the descriptive method because it’s not saying this is the set of something it’s describing the pattern not the set all right how about the set of natural numbers from two to four okay natural numbers indicates what type of number it is that make sure that we wouldn’t write things like 3.5 and it tells us where to start and to end so I think that that one that one is accurate there’s only one set of natural numbers from two to four and that’s the set we were given how about the set of integers between one and five do you remember what an integer is all right well integers are the natural numbers oops all right also zero and then all the negative natural numbers negative 3 negative 4 negative 5 etc okay so that the integers between one and five if you kind of look along this list find the ones between one and five and that is our set as well so that is another correct way to describe this set so in other words there is more than one way to correctly describe the set but there are also descriptions that are not sufficient to describe only that set all right how about the set containing red yellow and blue is it the set of primary colors a set of colors the set of colors in the flag of Romania or multiple options from those all right we’ll definitely part 8 a set of primary colors all right those are as long as you accept kind of that traditional definition of what a primary color is and that’s red yellow and blue all right a set of colors not complete enough we don’t know which colors we could write other sets that are not red yellow blue the set of colors in the flag of Romania all right well if you know this one or if you’ve looked it up the colors of the flag of Romania are red yellow and blue so that is a good description all right try it out use the descriptive method to describe the set B containing 2 4 6 8 10 and 12 okay so I would want to start by saying B is the set of all right and then we look at what type of number this is well those are natural numbers and they also happen to all be even okay so let’s use that type of number to describe this B is the set of even natural numbers and not all of them just a portion of them so we have to say where this starts and ends so you could say this a couple different ways you could say from 2 to 12 you could say between 1 and 13 or you

could even say between between other numbers like between how about 0 and 14 because the only even ones between 0 and 14 would be those two through 12 said okay how about the set negative 2 negative 1 0 1 2 3 4 5 6 all right well these are not just natural numbers because we have these negatives and 0 but you can use that term we just talked about an integer so this is the the set of integers okay let’s try it out from where to where or between where and where well if you do from it’s got to be from negative 2 to 6 or you could say the set of integers between negative 3 and 7 alright a third method to describe sets we’ve done listing a verbal description and then a more technical notation called set builder notation this one uses variables to represent different elements of a set it uses braces and then a vertical bar that’s red as such that okay so here’s this set containing 1 2 3 4 5 and 6 and this equals or is the same as here it is in set builder notation all right so if we walk through and read this this first X you always read this as the set of all X such that the set of all X and you can think of X just as like a general a general variable or a general general element in the set so it’s like seeing the set of all elements and the vertical bar means such that and then after the vertical bar we write we write the properties describe the elements in the set okay so the first part here let me erase this line okay this part all right there’s your new symbol for element and if you recall the end the fancy end there means natural numbers so we’re saying that X has to be a natural number so this is the set of all X such that X is a natural number and okay then the second part all right you might want to review inequality symbols if you don’t remember those but that means x is less than seven so we can say the set of all X such that X is a natural number less than seven all right you might be thinking why in the world would we use this what’s wrong with just listing one two three four five and six well there’s nothing wrong with it when it’s that simple but sometimes you end up with a set that’s very complicated and you need a shorthand way to describe it and there are situations where the set builder notation is a lot shorter and faster to read okay we won’t do a whole lot with this notation but it is used in your textbook example so I just want to make sure that you can read those sets and write some of the basic ones so that you can understand the examples alright so let’s try it out let’s write these sets in set builder notation and then read how how you would read the sets out loud okay the set our contains the elements two four and six so we want to write R equals and then we do use braces for the set builder notation and then you always start with that variable X and the vertical bar alright so it’s the set of all elements X such that and then we need to describe what elements we have so we need to come up with a symbolic way to describe two four and six all right so think about what type of numbers those are they are even they’re not decimals they’re not fractions all right well notice that they are all from that set e the even natural numbers so we can say this is the set of all X such that X is an element of e the even natural numbers all right so that narrows it down to two four six eight ten twelve etc but then we just we need to have a way to describe that we only want the first three to four and six all right well one way you could say it would be we have to make sure that X is less then you could

do seven you can do eight all right so we want even natural numbers less than seven okay and then we would read that out loud the set of all X such that X is an element of the even numbers and x is less than seven all right let’s set w contains the elements red yellow and blue all right so this is the set of all X such that and here well we don’t really have a mathematical way to describe those so we can kind of do a hybrid here and throw in a verbal description so we just want to say that X is all right and rather than just listing the colors we usually try to to describe it in some way so we could use the one we already talked about X is a primary color okay and the last one the set K contains the elements 10 12 14 16 18 so k equals the set of all X such that X is alright similar to the first example here x is an even natural number and right now we can’t just say less than we have to kind of cut this set off on both ends we want to skip 2 4 6 8 and we want to skip anything from 20 22 on up all right so we could say X is between two numbers and let’s say it that way x is between let’s do eight and twenty you could also do 9 and 19 or you could do eight and nineteen or nine and twenty any of those would work alright so for that first part the x is the element of right recall you might want to to use those symbols for natural numbers even natural numbers and odd natural numbers and I’ll throw this one in there in here as well there is a symbol for integers it’s a fancy Z and Z stands for zahlen that’s the German word for I think it’s I’m going to say it’s for a number i’ll have to check on that one but that’s from a german word ok let’s do one example trying all three of these all right designate the set s with elements 32 33 34 35 etc using these three different methods all right the roster method ok so we’ve we’ve been given the name of the set so we want to make sure we use that so s equals for this one they’ll it’s the roster or listing method so we use braces and then you just start listing these out and looks like I need a little more space all right and the set continue so you use the ellipses and then just end it in a bracket okay the descriptive method so we can say let’s say it this way s is the set of s is the set of all right well what do we have what seems to be the property of these numbers all right well you only have a few sets to really talk about right now you’ve got natural numbers even ones and odd ones or maybe integers all right so these are all natural numbers so let’s go with that s is the set of natural numbers all right well it continues on forever at the end there so we really just need to say where the set starts all right so we want to say you could say starting at 32 but it would be a little bit a little bit better to say may be greater than 31 it’sit’s better math vocabulary to say it that way this is the set of natural numbers greater than 31 all right you could also say greater than or equal to 32 but you probably wouldn’t want to say from 32 because then you have to save you know from where to where and it’s not really technically correct to say from 32 to infinity so stay away from that one okay and finally set builder notation s is the set of all always starts the same set of all X such that X is what we already know it’s a natural

number and we so that it’s greater than 31 x is greater than 31 alright writing a set using an ellipsis so sometimes we can also use an ellipsis in the middle of a set just to shorten the amount of writing we have to do so right the set containing all even natural numbers between 99 and 201 well there are quite a few of those and you probably don’t want to write all of those out so we start with the first one the first even natural number between those is a hundred and then you write just a couple until you kind of establish a pattern for what types of numbers are in this set usually three or four is good and actually let’s even skip that one three is good enough alright so we somebody looking at this can see that we mean the even natural numbers and then you write the ellipsis to indicate that it continues on in this same pattern for a while and then you just put the you know where the spot where you stop alright so the last one of those numbers is two hundred and then you don’t have to write all of those out all right using the roster method right the set of all letters of the English alphabet all right same thing just write a few until somebody gets the idea of what you mean okay another comma and then the last one all right and that’s it okay we’ll stop there and we’ll pick up in the next video with some more basic concepts from set theory but in this video we’ve talked about what a set is and then ways to three ways to mathematically describe sets