 Hello, and welcome to this very first screencast for Math 2.10, Communicating in Mathematics at Grand Valley State University. In this screencast, we're going to start at the ground level and discuss statements. So what is a statement? Well, in mathematics we define a statement to be a declarative sentence that is either true or false, but not both. That word declarative means that the sentence is actually declaring something, and it's not just a question or an exclamation. Here are some examples. January is the first month of the year. Now this is a sentence and it declares something, and it's definitely either true or false, but not both. Of course, in fact, this case it's true. July is the first month of the year, is also a statement, because it's a sentence that declares something, and it's definitely either true or false, but not both. Of course, the statement is false. I could pause here, just notice that statements above all have definite truth values, but also notice that the truth values depend pretty heavily upon understanding definitions. You probably assumed, like I would, that year means calendar year, but if I actually meant fiscal year like in business, then everything changes. The first statement about January being the first month of the quote-unquote year becomes false, and the second statement about July is now true. So while statements have definite truth values, just realize that we have to understand and agree upon definitions of terms before truth makes any sense. Let's see some more examples. Two is even. This is clearly a declarative statement that is definitely, in this case, true, although we do need to have a common understanding of what even means. We'll just go with what we, I think we all know at this point. There exist even numbers greater than 100 is a statement that is in this case definitely true. For example, 104. That's even, and it's greater than 100, so they exist. And here's one more. Every even number is greater than 100. That is definitely false because not every even number is greater than 100, for example, 2. So what does a non-statement look like? What time is it? That's a sentence, but it's not declarative. It's not claiming anything. It's just a question. So it's not a statement. Red is pretty. This is not a statement because although it's a declarative sentence, it's not definitely true or definitely false. It's a judgment call, and so it's not really suitable for mathematics. A more interesting non-statement is this one, 2x plus 10 equals 14. This is really a sentence, although there's no English in it, that says the expression 2x plus 10 on the left is equal to the number 14 on the right. It is declarative 2 since it's claiming that the two items are equal. But does it have a definite truth value? Well, it depends. It depends on what x is. If x is equal to 2, then the statement's true. But if x is equal to say 26, then it's false. Now since there's no single definite truth value for this sentence, the truth value depends on the value of x. This makes it not a statement. It's an equation, and obviously those are very important, but it's not a statement as such. Now let's move on to our very first concept check. Concept checks are quizzes that I'll be embedding inside these screencasts. I'll allow you to check and see how well you're understanding the concept being discussed. The way you should work these is listen to the statement of the question and the statement of the options, and then pause the video and think for a minute and make your selection of which one or ones seem to be most correct. And then unpause the video, and we'll have a quick discussion of the debrief. So for your first question, which of the following are statements? And you can just select all the ones that apply. x plus 2, x plus 2 equals 3. The sentence, the equation x plus 2 equals 3 has exactly one solution. Or the equation x plus 2 equals 3 has more than one solution. So again, pause the video and think for a minute and then come back. Okay, now that you had a chance to think, let's see which answers are correct here. The correct answers to this concept check are three and four only. One and two are not statements. Number one is not a statement because it's not a sentence at all. It's just an expression, x plus 2. It'll be like me coming up and saying green. That's not a sentence either. So x plus 2 by itself is what we would call an expression in mathematics, but not really a statement. x plus 2 equals 3 is a sentence, but it's not a statement for the reasons that we discussed on the last slide. The truth value of that sentence depends on what x is. For some values of x, that's true. For others, it isn't. And we don't want to be so indeterminate about things. Now three and four are correct because they are declarative statements and they are definitely true or definitely false. The first one says the equation x plus 2 equals 3 has exactly one solution. And that's certainly true. And the very fact that it's certainly true makes it a statement. In fact, x, of course, equals 1 is the only solution you have. Now the equation x plus 2 equals 3 has more than one solution. That is definitely false. And again, the very fact that it is definitely false makes it a statement. The equation x plus 2 equals 3 does not have more than one solution. It just has the 1 at x equals 1. So let's review what we've learned in this video. We've learned that statements are declarative sentences that are either true or false, but not both. And sentences that are not declarative, like what time is it, are not considered statements. And importantly, statements whose truth value depends on the value of a variable are not statements either. But sometimes we can quantify the variable using, say, there exists a solution to this equation that can turn that expression into a statement. That's all. Thanks for watching.