# 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