MATH 110 Sec 2.1 (F2019): Basic Properties of Sets

Today, we’ll begin our study of chapter 2 by talking about the basic properties of sets Georg Cantor, a German mathematician who was actually born in St Petersburg, Russia in 1845, is considered to be the father of set theory In the beginning, many of his ideas were highly controversial And he experienced strong resistance to his ideas from fellow mathematicians Ultimately, his ideas prevailed, though And in the process, he created the field called set theory One of the ideas that Cantor had to deal with was the concept of infinity Going back to the idea of a set, any group or collection of objects is called a set The objects that belong to the set are its elements or members For example, the set consisting of the four seasons has spring, summer, fall, and winter as its elements There are two common methods for writing sets The first method is just describing the set in words The second method is listing the elements of the set inside a pair of braces and using commas to separate the elements And this is called the roster method There is a third method that we’ll talk about later But these are the ones that we’ll start off with For example, word description followed by the roster method for the set of denominations of US currency and production at this time I just did a word description It describes a set In the roster method, you would actually list every bill of US currency that’s currently in production There’s a $1 bill, a $5 bill, a $10 bill, a $20 bill, a $50 bill, and a $100 bill You would write them down with commas separating them inside of a pair of brackets And that’s called the roster method The set of US states boarding the Pacific That’s the word description of the set The roster method would be the list each state individually separated by commas inside of a set of brackets And they are California, Oregon, Washington, Alaska, and Hawaii So the ideas are simple You can describe the set in words, or you can list the elements using the roster method Here’s an exercise Use the roster method to represent the set of days of the week Again, you simply list them with commas inside of braces– Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday Notice that I put them in a certain order But in a set, order it is not important So had I scramble those names up, it would not have mattered It would still be the same set How about this one? Write a word description for the set, capital A equals, and then, inside of the braces, you get a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z How would you describe that? Well, I would say A is a set of letters of the English alphabet And I say the English alphabet because there are other alphabets besides the English alphabet And they’re not all the same Because is a math course, we’ll be particularly interested in certain sets And I’m going to mention a few of them now And they’re used extensively in many areas of mathematics First of all, we have the natural numbers, sometimes called the counting numbers We use a capital N when we’re talking about those numbers And it’s simply the numbers we count with– 1, 2, 3, 4, 5 And the ellipsis, the dot dot dot at the end, just means that the pattern continues forevermore The whole numbers, we use a capital W to represent that And it is the same as the counting numbers with the exception of an addition of 0 So the whole numbers begin with 0 and continue through the counting numbers The natural numbers, or counting numbers, do not include 0 Then, we have the integers We use capital I for that In the integers, if you look at the middle at the 0, to the right, you get the counting numbers To the left of 0, you get the negatives of the counting numbers And notice the ellipsis, the dot dot dot in both directions And that means the pattern continues infinitely in both directions So the integers consist of 0 and, to the right, 1, 2, 3, 4, 5, the counting numbers, and, to the left, the negative of the counting numbers, negative 1, negative 2, negative 3, negative 4, and on and on You might notice that the counting numbers are inside the set of integers So counting numbers are also integers So you could also say that the positive integers are the same thing as the counting numbers There are a lot of little relationships like that that you could notice if you really wanted to

The rational numbers, we use Q for that The rational numbers is the set of fractions, which are formed as an integer divided by another integer as long as you’re not dividing by 0 You can also define Q as a set of all terminating or repeating decimals And it turns out, those two definitions are equivalent The irrational numbers, we use a script I A regular I for integers, a script I for irrational numbers That’s the set of all decimals that don’t terminate and don’t repeat And finally, the set of real numbers, we use capital R. That’s the set of all rational or irrational If you throw the rational numbers and irrational numbers together, you get the set of real numbers And I also say that, having defined real numbers, you can also go back and define the irrational numbers as the set of real numbers that are not rational You couldn’t really define it that way until I talked about what rational numbers are and irrational numbers are But once I’ve defined those terms, I can also talk about irrational numbers as real numbers that are not rational Those are the kind of things that you’ll need to have some familiarity with so that, if a question uses those names, you’ll know what it means How about this? Use the roster method to write each of the given sets The set of natural numbers less than 5 Remember, the natural numbers are the counting numbers, 1, 2, 3, 4, 5, 6, 7, and on and on and on If you only want the ones less than 5, you’ve got to mark out 5 and everything to the right So the natural numbers less than 5 consist of the set of elements 1, 2, 3, and 4 How about this? The solution set of the equation x plus 5 is equal to minus 1 If you take that equation and solve for x by subtracting 5 from both sides, you’ll find out that x is equal to minus 6 So the solution set is the set containing the single element negative 6 And this one The set of negative integers greater than minus 4 Remember, the negative integers are part of the integers The integers– if you think of 0 as sort of being in the middle to the right, you get the counting numbers, 1, 2, 3, 4, 5, and on and on To the left, you get minus 1, minus 2, minus 3, and so on to the right If you’re looking for the negative integers greater than minus 4, remember, the negative integers, if you’re moving to the right, stop at minus 1 Once you get to the right of minus 1, you’re not in the negative integers anymore But you only want the ones greater than minus 4 You also don’t want to go to the left of minus 4 with things like minus 5, minus 6, and so on So the set of negative integers greater than minus 4 has to be to the right of minus 4 but stop once you get to the last negative integer, which is minus 1, or the last negative integer before you get to 0 And remember, it didn’t say greater than or equal to minus 4 It just said greater than minus 4 There are only three of them The solution set is to set consisting of minus 3, minus 2, and minus 1 There’s also a term called well defined when we’re talking about sets A set is well defined if it’s possible to determine whether any given item is an element of the set And the best way to illustrate it is through some simple examples If I said, the set of letters of the English alphabet is well defined, you should say, yes, that’s right, because I can tell you whether anything that you flash in front of me is in that set or not But if I say, the set of great songs is well defined, you might pause with that, because the set of great songs, how do you decide if a song is in it or not Because what you think is great might not be what I think is great And in fact, you might think one time that a song is great And a year later, you might think it’s not That’s sort of just an opinion It’s not well defined And that is a sense of what well defined means If you can nail it down for sure that you can look at something and decide if it’s in the set or not, that set is well defined If there’s any ambiguity, then it’s not There’s a lot of notation in mathematics And there’s a lot of notation in set theory If I give you the statement, 4 is an element of the set of natural numbers, a mathematician is going to want to write that in a more condensed format We use a lot of symbols We use the symbol– sometimes, I call that the pitchfork symbol And it means, “is an element of.” I would write 4 and then write the symbol for “is an element of.” And capital N is the letter we use for natural numbers

And I would read that symbol as “is an element of.” If I wanted to state that something is not an element of a set, we take that same symbol and put a slash through it So in order to write, minus 3 is not an element of the set of natural numbers, I would take the symbol for “is an element of” and put a slash through it And I would say, minus 3 is not an element, by putting a slash through there, of the set capital N, which represents the set of natural numbers So it’s standard to put a slash through a symbol to mean not Determine whether each statement is true or false a says 4 is an element of the set containing the elements 2, 3, 4, and 7 It’s true if there is a 4 in that set And there is So it’s true b– minus 5 is an element of the natural numbers Remember, the natural numbers are the counting numbers Negative 5 is not a counting number If you look in that set, you don’t see negative 5 So that’s false c says, one half is not an integer, is not an element of the set of integers Remember, the integers have 0 and then, to the right, the counting numbers and, to the left, the negative counting numbers One half is not in there at all If you look, there is no one half in there So it’s true that it’s not an element You’ve got to be careful That symbol says “not an element of.” So it’s true that one half is not an integer And finally, d– the set of nice cars is a well-defined set Is that true or false? Well, again, very similar to what I said earlier with the example about songs, that’s sort of an opinion question It is not well defined So the set of nice cars is definitely not a well-defined set We also talk about the empty set Or sometimes, we say the null set It’s the set that contains no elements We use two different symbols pretty regularly to stand for the empty set And that’s a circle with a slash through it or a pair of braces with nothing inside You’ll see both of those symbols So if you see either one, you’ll know that we’re talking about the empty set As an example of the empty set, consider the set of natural numbers that are negative integers or the set of birds that live at the bottom of the ocean There simply aren’t any natural numbers that are negative because natural numbers are counting numbers There are no birds, as far as I know, that live at the bottom of the ocean Those will both be simple examples of empty sets Now, I talked about earlier there were two common methods of writing sets And that is to do it in words or to use the roster method There’s actually a third method, which, in a sense, is a combination of the two It’s called set-builder notation Set-builder notation is especially useful when you describe infinite sets or things that are hard to write out as well In set-builder notation, the set of natural numbers greater than 7 is written as follows You put your braces on the left and right And then, you write– and you don’t always have to use the letter x, but it’s quite common– x, then a vertical line followed by, x is an element of capital N, which is the natural numbers And at the same time, x is greater than 7 So this is sort of a combination between writing it in words and writing it out element by element You’re using words But you’re also using components of the roster method, in particular, the braces Here’s how you should read this The opening left brace should be read as “the set.” And the x should be read “of all elements x.” So you’re reading left brace x as “the set of all elements x.” The vertical line is read as “such that.” Then, you’ve got the next part, which says, x is an element of the natural numbers, or x is an element of the set of natural numbers And x is greater than 7 So you move across from left to right The opening brace is read, “the set.” The x is read as “of the set of all elements x.” The vertical line is read “such that.” And you keep reading across It’s simply impossible to list all elements of the set because there are infinite number of them But set-builder notation precisely defines the set by describing its elements For example, let’s use set-builder notation to write the following sets The set of integers greater than minus 3 So you could write x and use the symbol for “is an element of.” Capital I is integer So x is an element of the integers In other words, x is an integer and is greater than minus 3

It takes a little practice But once you get used to it, you simply read that out to yourself And you’ve got an idea of what the set really is And another one The set of whole numbers less than 1,000 So you start writing out, opening brace, x, vertical line That’s the set of all x such that x is a whole number Remember, capital W stands for the set of whole numbers You write x with the symbol for “is an element of.” x is an element of the whole numbers And x is also less than 1,000 Practice makes perfect here Practice will help you see, if you’ve done a few, how it works Another definition A set is finite if the number of elements in the set is a whole number For example, if I said, the set containing the elements 1, 3, 6, and 7 is blank– and I want to put finite if it’s true or not finite if it’s false– because it has four elements And 4 is a whole number Remember, a set is finite if the number of elements in the set is a whole number Well, it does have four elements And 4 is a whole number A finite set, you can count how many elements are in there And you say whole number instead of counting number because you could have the empty set, which would have zero elements Conversely, if I take the set containing 1, 3, 5, dot, dot, dot, which means it continues forever, you never get to an end So you can’t count them and get a whole number answer So that means that set is not finite because it has an infinity of elements And infinity is not a whole number Another definition The cardinal number of a finite set is just the number of elements in the set It’s just a fancy way to say how many elements are in the set The cardinal number of a finite set is denoted n of A. Little n and, in parentheses, the name of the set So that would be read “n of A.” So the cardinal number of a finite set is the number of elements in the set For instance, if the set A is the set containing 1, 3, 6, and 7, then the number of elements in A is 4 Another way of saying that is, the cardinal number of the set A is 4 You could also say that A has a cardinality of 4 Let’s find the cardinality of each of these sets J is a set containing 2 and 5 Well, that’s easy enough There are only two elements in that set So the cardinality is 2 You could also write, n of J is equal to 2 Now, before I do the next one, I want to mention something that’s useful in answering this next question There is a formula for counting elements in a big set if the elements are consecutive integers So you can kind of guess the next example I’m going to do is with consecutive integers If you had a set containing the element 7, 8, 9, dot, dot, dot, all the way to 22 or something like 4, 5, 6, 7, 8, 9, 10, all the way up to 45, there’s an easy way to count those And that is to use this formula The number of elements in a set of consecutive integers is simply the biggest number minus the smallest number plus 1 So for that 7, 8, 9, dot, dot, dot, all way to 22, you would take 22 minus 7, the biggest one minus the smallest one, and add 1 And that would come out to be 16 If you don’t believe me, count them out and see There are actually 16 elements in that set If I took the 4, 5, 6, dot, dot, dot, to 45, it would be 45 minus 4 plus 1 And that comes out to 42 This only works for consecutive integers, though If they’re not consecutive, you can’t use this formula, with one exception Sometimes, you have a set with a pattern that can be shown to have the same number of elements as another set that does have consecutive integers For example, if I asked you to find the cardinality of the set 6, 9, 12, dot, dot, dot, all the way to 33, those are all multiples of 3 6, 9, 12, 15, they’re all divisible by 3 So those are all multiples of 3 all the way to 33 Those aren’t consecutive So you’d say, well, you can’t use that formula above because they’re not consecutive And that’s true enough But if the numbers have a certain pattern, you might be able to associate that set with another set that is consecutive that has the same number of elements And the trick of that is try dividing every element by 3 and look what happens If you divide 6 by 3, you get 2 If you divide 9 by 3, you get 3 If you divide 12 by 3, you get 4 If you divide 15 by 3, you get 5, all the way to the end And look, those numbers are consecutive–

2, 3, 4, 5, dot, dot, dot, 11 And they have the same number of elements as the original set because all you’re doing is dividing everyone by 3 You’re not changing how many there are But now that you know that 2, 3, 4, 5, dot, dot, dot, to 11 has the same number of elements and they are consecutive, I can use the formula for the consecutive one– biggest minus smallest plus 1 That’s 11 minus 2 plus 1, which is 10 So if the set containing 2, 3, 4, 5, all the way to 11 has 10, then the one that I was really looking at originally must also have 10 So it’s sort of a trick there when you have multiples If you divide out the multiples, you get consecutive integers And you can use the formula on those You can kind of guess the next example I want to work uses that idea So the next example is the set S containing 3, 4, 5, 6, dot, dot, dot, 31 Those are consecutive So you can say, 31 minus 3, which is 28, plus 1 is 29 And you could also write n of S is equal to 29 The cardinality of that set is 29 Now, this one– the set containing 3 comma 3 comma 7 comma 21– that’s basically a trick question You really should not list the same element more than once in a set But if you do, you don’t count it more than once So even though that 3 got listed twice, 3, 7, and 21 are the only elements of that set Just be on your toes Occasionally, they’ll throw something in like that to make you think But if an element is written down more than once, it’s only counted once So the cardinality of that set is 3 And you could say n of T is equal to 3 We also talk about equal sets A set A is said to be equal to a set B And you actually use an equal sign You denote that A equals B if, and only if, A and B had exactly the same elements So if I gave you the set containing D, E, and F, and asked you to compare it to the set containing E, F, D, you could say that they’re equal because they have the same elements There’s a D in both, an E in both, and an F in both And there’s nothing that’s in one and not the other Remember, the order of sets does not matter But if I asked you to compare the set containing D, E, and F to the set containing D, E, and T, those sets would not be equal because there’s a D in both, there’s an E in both, but there’s an F in one and not in the other And there’s a T in one and not the other So those sets are not equal They have to have the exact number of elements And they have to be the same elements There’s a looser term called equivalence Set A is said to be equivalent to set B And instead of putting an equal sign, you put a tilde or sort of a squiggle That represents equivalence A is equivalent to B if, and only if, A and B had the same number of elements If you use that notation we developed earlier, you can also say A is equivalent to B, provided that the number of elements in A is equal to the number of elements in B. In other words, you’re simply counting You don’t care what the elements are, there just have to be the same number in both Equality, they have to be exactly the same elements Equivalence, they just have to have the same number of elements So for instance, the set containing D, E, and F is equivalent to the set containing E, F, and D You could have used equal there because they have exactly the same element, but they’re also equivalent But the other one, D, E, and F compared to D, E, and T, they’re not equal because one’s got an F and the other has got a T But they are equivalent because there are three elements on the left, D, E, F And there are three elements on the right, D, E, T So you’re simply counting when you’re looking at equivalence Get off to a good start in this course Go ahead and get your homework done as soon as possible Not only will get a bonus for finishing early But if the material is still fresh in your mind when you do this and you’ve got plenty of time to get help if you do run into a problem, that’ll give you a better chance of getting these things done correctly Also, I would say you’ll learn a lot more if you’re an active learner This may take a little bit more of your upfront time But it’ll pay off in the end if you’ll become an active learner For example, I know there’s a problem in the homework that refers to vowels Now, I haven’t mentioned that term But if you’ve forgotten your vowels, allow a little bit extra time and go online and refresh your memory

And that’s what I mean by being an active learner Take some of the load onto yourself Because if you learn something yourself, you’re better able to remember it There are also some problems where you need to read a simple graph or some kind of table to answer the question Again, it’s easy But you may need a little extra time to closely examine the graph of the table This is part of your responsibility as a student in this course And it’s really important because, believe it or not, active learning really does help you to better retain what you learn And you’re going to be better off for having done this stuff So good luck on your homework Remember, I’m here to help you If you need to send me an email, most of the time, I’ll get back you very, very quickly You can also make an appointment to see me one on one or just ask a question Help is also available at the MTLC and other campus units I have some office hours over there myself And even if I’m not there, there are others there that can help you during the open tutoring hours So go get your homework done