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

Welcome back. In this problem, we'd like to translate the following conjunctions from words to symbols, then we'll determine the truth value. And we've been given the simple statements for P, Q, and R. Now, the conjunction that we'd like to translate is some courses don't give tests and give homework. So, let's just look piece by piece. Our first piece here is some courses don't give tests. Now, looking at our simple statements, the only simple statement that talks about tests is statement P. All courses give tests. Now, what we have in our conjunction is some courses don't give tests. This is the opposite of saying all courses give tests. If it's not true that all courses give tests, that means at least some courses don't give tests. Because this piece is the opposite of the simple statement P, we know that this piece is not P. Next, we have the word and, which we can represent with this symbol here for the word and. Finally, we have give homework. Now, this is a sentence fragment. It doesn't have a subject, and sometimes we see this with compound statements. See, when we combine two simple statements together with the word and, and both of the simple statements have the same subject, we don't actually have to repeat the subject of the second statement. So here we can assume that some courses is the subject for both of the simple statements we combined. This statement is equivalent to saying some courses don't give tests, and some courses give homework. We just don't need to repeat some courses. Now, some courses give homework is simple statement Q. So this next piece here is statement Q, and symbolically, we can represent this as not P and Q. Now that we have it from words to symbols, let's determine the truth value. And I will say oftentimes it is easier to determine the truth value from the version in words. So we'll be going back to that version in words. But as a reminder for conjunctions, we need both of the simple statements we're combining together to be true individually. So let's check each of them individually. Now, some courses don't give tests. This is definitely true. I have been in courses before that didn't give tests, and I'm sure maybe you have too. Maybe they gave essays or projects, but it's definitely true that there are some courses that don't give tests. Next, we have some courses give homework. Well, this is definitely true, because I have been in many courses, and I'm sure you have too, that did give homework. So we can see that our conjunction is combining two statements that are true on their own, which means the conjunction itself is also true. It is true that some courses don't give tests and give homework, great work.