 Hello, and welcome to another screencast where we're going to be using predicates and propositional functions to translate from logic back and forth to English. Except this time we're going backwards. We're starting with some English statements about system requirements, and then translating to logic. Now, what we're about to see here has many different possible solutions. All of these have to do with quantifying certain predicates. Remember a predicate is just a function where we put in a specific instance of a variable and get a proposition out that's either true or false. All of these depend on how we choose the predicates, and so we have to be very careful and explicit about that. This is just what you're going to see here in the screencast is just one possible way to look at these statements. There are others as long as they're consistent with your choice of predicates, and we'll see maybe further what we mean as we go. So here's a statement about a system. Every user has access to an electronic mailbox. Now, first of all, I do note the word every here, and that sounds like a universal quantification, and we'll make that happen in just a minute. But we decided on a predicate first of all, and so I'm going to say m of x is the statement x can access a mailbox. x can access a mailbox. There are different ways to state this, and maybe I'll give an alternative in just a minute. But if you let m of x be the predicate, x can access a mailbox. Again, this statement is, this is not really a statement at all. This is just a propositional function, and it could be true or false depending on who x is. So I need to quantify it to make this a real proposition. So we see every user, that sounds like a universal quantification, okay? And so this English statement would translate into every user for all x's, those users can access a mailbox. And that is one correct way to write down this logical statement, this English statement up here if I use this predicate. Another way to do this would be to define a predicate a little bit more broadly, like for example, m of x, y equals x can access system y. Make it a little bit more broad. And in that case the way we would write this sentence out is for every x, m x mailbox, okay? Every user can access a mailbox. Both of these are considered correct because they are consistent with the way that we've defined the predicate m. So there's more than one right answer for these kinds of questions here. So here's another English statement. The system mailbox can be accessed by everyone in the group if the file system is locked. Now we do have again we have some choices as to how our predicates go but we don't get too much choice in terms of how the logical sentence is set up. We focus in on the word if. So this is going to be a conditional statement and there's sort of an implied then over here if you wanted to rewrite this in reverse order, it would say if the file system is locked then the system mailbox can be accessed by everyone in the group. So we're going to go ahead and put an arrow here because this is going to be a conditional statement. So the if part here's where we have to define our predicate. And again there's more than one way to do this. So we could say let's let L of y equal the statement system y is locked. System y is locked. Okay? So in that case we could just simply say this is not necessarily a quantified statement. We could just say L of file system other things in a computer system could be locked not only the file system. So if the file system is locked then everyone in the group can access the system mailbox. So in the previous example we had let m of x equal the statement x can access a mailbox. And if we keep that notation, keep that way of describing m, then the conclusion of this if then statement says the system mailbox can be accessed by everyone. That would just be saying everyone has access to the mailbox. Okay? So that would be a correct way, one correct way, not the only correct way to write this English statement in terms of logical symbols. Here's a third statement. The firewall is in a diagnostic state if only if sorry the proxy server is in a diagnostic state. Clearly both of these statements have to do with something being in a diagnostic state. So that's how I'm going to define my predicate here. Let's define the predicate say D of y to mean system y is in a diagnostic state. System y is in a diagnostic state. Again this is not the only way diagnostic this is not the only way we could define a predicate in this case. We just have to define it and be clear about it and be consistent. So only if means the same thing as if. So if the proxy server is in a diagnostic state then the firewall is in a diagnostic state. So we could just simply write this as D proxy server implies D firewall. And again just to reiterate this there's not only one way to write this but whatever however you write it you must be very clear about what you mean by your predicates and then stay consistent with your notation. Let's move on to this last one here. At least one router is functioning normally if focus on the if the throughput is between 100 kilobits per second and 500 kilobits per second and another logical word the proxy server is not in diagnostic mode. So we're going to use the same D of y predicate for something being in diagnostic mode. But we have a couple of other new situations functioning normally. Let's just call n of y. Let that be the predicate y is functioning normally functioning normally. And then I have something about throughput. So you do not have to know what throughput means or kbps means but I'll let t of ab that sounds like a two variable function here t of ab to mean the throughput is between a and b. So with those three predicates defined this becomes not so bad. So we have an if then statement. Here's the if part. If the throughput is between 100 and 500 and the proxy server is not in diagnostic mode. So not diagnostic mode proxy server. That's the if part. Then something happens. At least one router at least one means there exists. So there exists a router. And we can even fix this in our predicate. It seems like what should you put here y or router. That depends on what the domain of this propositional function is. Let's just define the domain to be the set of all routers. So functioning normally for my definition is going to only apply to routers. So there would exist at least one router such that that router is functioning normally. With those choices of predicates and domains this statement would be correct. And again there are others. It depends on how you define your predicates and what your domains come out to be. So this is definitely a situation where there's more than one right answer to this question. Good luck with it and field your questions to your instructor.