 Hello and welcome to the screencast on Open Sentences. So I want to take you back all the way to the very first screencast in this whole series, screencast 111, and we were discussing statements versus sentences that weren't statements. And this equation right here is an example of latter. It's a sentence that is not a statement. Now why is this not a statement? Well remember a statement is a well-formed declarative English sentence, which this is, but it has a definite true or false value to it. You can look at it and tell that it's true or false. This one we cannot determine the truth or falsehood value of the statement or the sentence because of the variable. The fact that there's an x right here means that this sentence 2x plus 10 equals 14 is sometimes true, but sometimes false as well. If I put in x equals 2, then I have a true statement. If I put in x equals 3, I have a false statement. The universal set for this variable refers to just the set from which these variable values come. Here I was putting in whole numbers, but really you could set your universal set to be anything you want, depending on context. Let's suppose that the universal set for now is just the set of all real numbers. Now so we have a statement that is not a sentence because of the variables. Here's another example of such a thing, except now we have two variables, an x and a y. And again, this is a sentence. It's well formed. It's well understood in math or you could even say it in English, but it's not a statement because we can't tell what the truth value is until we know an x and a y value. There are a lot of x values and y values that make this true. We have to think of them in pairs, right? Just to say x equals 5 doesn't really quantify this enough. I'd have to say x equals 5, y equals 4. And that, if you plug those in, would give you a true statement. 5 minus 4 is indeed equal to 1. But if I have x equals 5 and say y equals 2, then that does not give me a true statement. So there's another example of a sentence that's almost a statement. If you could just plug in the holes, plug in numbers for the variables, you'd have yourself an actual statement that's definitely true or definitely false. So we have a name for such things and that is an open sentence. Okay, that is a sentence. Could be an English sentence. It could be a mathematical expression like an equation that involves variables and it's such that once you put in specific values for the variables, whether there's one of them or two of them or many of them, then what you get is a statement. Something that's definitely true or definitely false like our two examples over here. We use the word open to specify the fact that we have some holes to plug in here. Once I put in a number for x, then maybe I can tell whether this is a true statement or not. But until I do, it's ambiguous and it's merely an open sentence. So let's look at a couple of other open sentences here. 2x plus 10 equals 14 and the inequality x squared less than or equal to four and I want to ask the same question both times. What is the set of variable values that makes this open sentence into a true statement? We saw that I could put in certain values for x that might make this a false statement, but what are the variable values that make it a true statement? Here I'm going to fix in this first bullet point, this first example, I'm going to fix down the universal set to be the set of real numbers just to open up my options completely. So what is the set of all real numbers that makes the equation 2x plus 10 equal 14 true? Well, that's fairly easy. We can just do a little math in our heads and realize that the set consisting of the number two is the set of all real numbers that makes this a true statement. There are no others. There's only this one here and I'm putting it in my curly brackets. By the same token, if I look at the inequality x squared less than or equal to four, what is the set of all variable values that make this true? Now notice here I'm changing the universal set to be integers. So I want to only think about whole numbers, positive negative zero whole numbers. What is the set of all integers that makes this inequality true? Well, with a little bit of thinking, I can start my curly brackets. I need stuff that's close to four and we'll find that negative two works because negative two squared is four. That's less than or equal to four. Negative one works. So does zero. So does one. So does two. And that's it. So that's the complete list of integers that makes this inequality true. So what we're doing here is we're taking an open sentence and looking for the variable values from the universal set that makes the open sentence into not just a statement but a true statement. That set is generally known as the truth set of an open sentence. Again, just the collection of objects from the universal set that can be substituted in for the variable to make the sentence a true statement. So let's end off with a concept check to see how well you understand the notions of open sentences and truth sets. So suppose that the universal set is the set of integers, the set of all whole numbers both positive, negative, and zero. And we're going to look at the open sentence cube root of n is an integer. Okay, so again n is an integer but here's an open sentence cube root of n is an integer. So this is an open sentence because it has this variable in it and the variable is some other integer. So what's the truth set? What is the set of all integers that I could plug in to make this quoted sentence a true statement? Is that set empty? Are there no integers that work? Is it the set of cube roots that you see here? Is it the set of plus or minus cube roots that you see here? Is it the set 0, 1, 8, 27, 64, and so forth? Is it the set 0, plus or minus 1, plus or minus 8, and so forth? Or is it basically everything in the universal set, the entire set z of integers? So think about this one and play around with the problem and pause the video and when you come back we'll reveal the answer. So the answer here, what is the set of all integers n such that the cube root of n is an integer would be this one here in e. Now again why is that true? Here are the inputs. This is where it would be going in to in. So first of all the sentence is claiming that the cube root of n is an integer. So these guys here if I take their cube roots those are not going to be integers. So these guys are out. I know that 0, the cube root of 0 is 0, the cube root of 1 is 1, the cube root of 8 is 2, the cube root of 27 is 3, the cube root of 64 is 4. So all these guys work. That makes empty out of the question. But also if you remember your basic math here the cube root of a negative number is okay to compute so I also want to include negative 8 for example. The cube root of negative 8 is negative 2 and that's an integer. So all these guys work but not the entire set z. You can easily try out say n equal to 5. If I look at the cube root of 5 just a quick punch through a calculator will tell me that this is not it is not an integer. So it's not everything. It's some set halfway in between not halfway really but empty and between empty and the entire set of integers is actually this set right here. Those are all the integers which when evaluated into n make this quoted thing a true statement. So we'll have many many opportunities to return to the notion of open sentences and true sets later on. But for now thanks for watching.