Translate the following conjunctions from words to symbols. Then, determine the truth value. p p : All courses give tests. q q : Some courses give homework. r r : Some courses have textbooks. All courses give tests, some courses give homework, and all courses don’t have textbooks.

Welcome back. In this problem, we'd like to translate the following conjunction. All courses give tests, some courses give homework, and all courses don't have textbooks from words to symbols, and then determine the truth value. Now, let's just go piece by piece. Our first piece is all courses give tests, which is exactly what simple statement P is. So let's start with a P. Next, we have a comma, let's leave that for now and keep going, but we'll come back to it. Then we have some courses give homework. Now, that's exactly what simple statement Q is. Some courses give homework. So let's put a Q here. Then we have another comma, and we have the word and, so we can add this symbol for the word and, and then we have all courses don't have textbooks. Now, the only statement about textbooks is statement R. Some courses have textbooks, so it has to be related to statement R, and actually we can see that they're opposites. The opposite of some courses having textbooks would be if no courses had textbooks, or if all courses don't have textbooks. So what we have in our conjunction is the opposite of R or not R. Now, one thing we saw when translating from symbols to words in a previous practice, was that when we're listing things in English, we don't have to keep repeating the word and. We don't have to say this and this and this and this. We'll often just remove those ands up until we're adding the last thing and replace them with commas to avoid repetition. And that's what's happening here. The statement that we have is the exact same thing as all courses give tests, and some courses give homework, and all courses don't have textbooks. We just don't need to repeat the word and in English. We do need to add that extra symbol for and when we're representing this symbolically. So we do need to put in that and symbol between P and Q. Now that we've represented it symbolically, let's determine the truth value, and we will be using the version in words because that's often easier. Now, when we're looking at conjunctions, they are true when all of the statements we're combining together are true on their own. So now we need to check all three of our statements individually to see if they're true, and the conjunction will only be true if all three of our pieces are true on their own. Now, our first piece is all courses give tests. I have definitely been in courses that instead give papers or projects in place of tests, so this is going to be a false statement. Then we have some courses give homework, that's true. I've been in courses that do give homework, and we have all courses don't have textbooks. Well, this is false because I've definitely been in courses that have textbooks. We can see that we're combining two false statements and a true statement, so they are not all true, meaning that our conjunction can't be true either. So our conjunction is false, and we can say that P and Q and not R is false. Great work.