 Hello, and welcome to a screencast where we're going to do some translation from logical statements into English. These all involve a propositional function that has two variables and we're quantifying it and just wanting to know how do we get a sensible English statement out of these logical symbols. So, and this is actually taken from exercise 38 in section 1.3 of the book Discrete Mathematics and its applications by Kenneth Rosen that we use for Math 225. So here in these five statements here SXY is a propositional function that has to do with systems like computer systems and states that those systems may be in. SXY means that X is in state Y. And so, S has a built in English translation to it. We just need to put it together with logical symbols and make some sense out of it. So let's begin with looking at the first statement here. The first statement would say this part is simple enough. It would be saying there exists something. Okay, there exists an X. I changed my pen to blue. There exists an X such that S of X open is a true statement. So let's go down and see if we can make sense of this. There exists an X such that system X is open. And that would be a pretty good English translation. We could just say there exists a system that is open. That would be one way to say that. So simple as that. Just simply translating piece by piece. Now with that let's move on to the second one. This is a universal quantifier. So there for all systems X. I'm looking in here and seeing the disjunction here. So for all systems X, SX malfunctioning is true or, remember that's what this means, or SX diagnostic is true. So one way we can translate the for all X is just merely to say every. That's one acceptable translation of for all. So we could say, one way to phrase this is to say that every system is, we can throw in the word either to make it make a little bit more English sense is either malfunctioning or in a diagnostic state. So every system is either malfunctioning or diagnostic. Now the third here, there's another disjunction but it's between two quantified statements. So it's this or this. So let's do each one of those individually. The first statement is exactly what we did in number one. So they would say there exists a system that is open. And then the second one is similar. There exists a system that is diagnostic. And now the way we could translate that is just to simply take those two statements and join them with an or. So we could say that one acceptable way to translate their exists is sum. So we could say some systems are open comma or some systems are diagnostic. We should emphasize that we don't have to know anything about systems diagnostic or anything relating to the terms themselves in these statements. We're just merely going from one logical form to a language form. Fourth one here would say there exists an x such that not s is available x is available. Now it's kind of an awkward way to say it. So one way we could rephrase that is to say that there exists a system. Instead of saying not is available, we could just say that is unavailable. There exists a system that is unavailable. That's what not x available means. And finally this one here is pretty similar. This is a for all. So for all x not sx working. That means that for all systems x, x is not working. And so we could say that or we could say none of these systems are working. Notice that there are multiple ways to phrase these, but they all capture the same logical essence here going by the quantifier and what the meaning of the propositional function and whatever other logical symbols may be involved. Thanks for watching.