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. It is not true that some courses have textbooks and give homework.

Welcome back. In this problem, we'd like to translate the following statement. It is not true that some courses have textbooks and give homework from words to symbols, then we'll determine the truth value. Now, we start off the statement with, it is not true that. So that means that whatever comes next, we're going to take that and get the negation. It is not true that is what we put in front of a statement to get its negation. So, this part is signifying a negation. So let's figure out what it is that we're actually negating, and then we'll put that not symbol in front of it. So we're gonna come back to that. Now here we have some courses have textbooks. Now, this is exactly the same as statement R, so we have statement R here. Then we have the word and, so we're going to want to include our symbol for a conjunction. And then we have give homework. Now again, this is a sentence fragment, so that means that we have the same subject that we had in the first part of the conjunction, some courses. So this is some courses give homework, and again, we just don't repeat in English those repeated subjects. So that's why we have just the fragment here of give homework. Now, some courses give homework is exactly the same as the simple statement queue, so we're gonna add our cue here. So what this part. R and Q is representing is the piece in front of the phrase, it is not true that. Now, again, it is not true that it is going to negate the following piece, so we want to negate this whole conjunction, meaning that we want to put it in parentheses, and then put the negation symbol outside, saying that we're negating the whole conjunction and not just the simple statement R. Now that we have it represented in symbols, we want to determine the truth value. Now, we want to see the truth value of the negation of R and Q. So one of the easiest ways to do this is to find the truth value of R and Q, the conjunction piece, and then take the opposite truth value for its negation, which is what we have here. So that's what we're going to do. Let's start by seeing if R and Q is true. Now, again, a conjunction is true if all of the simple statements we're combining together are true on their own. So we're gonna check if R is true and if Q is true to see if R and Q is true. Now, R is the simple statement some courses have textbooks, and that is true. I have been in courses with textbooks, and I'm sure you have too. Now the next piece is some courses give homework, and again, this is true. I've been in courses with homework and I'm sure you have been too. So because both R and Q are true individually, that means that the conjunction R and Q is true. So, when we take the negation of that conjunction, we end up getting a false statement because the negation has the opposite truth value. So we can say that it is false, that it is not true that some courses have textbooks and give homework. That was a tricky one, so great job.