Translate the following conjunctions from symbols to words. Then, determine the truth value. p p : Ten is a whole number. q q : Five is less than three. r r : Two times seven is fourteen. ~ ( p Λ q ) (p Λ q)

Welcome back. In this problem, we'd like to translate this statement here from symbols to words, and we've been given the simple statements for P, Q, and R to help us out. Then we'd like to determine the truth value. Now, here we have not P and Q. This is one of the first statements we've ever seen with parentheses. Now parentheses are going to work here much like they do in other areas of math. We want to do the thing inside the parentheses first. So essentially this is saying take P and Q, the conjunction, and find the negation of that conjunction. Now, because we want to start in the parentheses, we're going to start with P, and P is the simple statement, 10 is a whole number. So let's write that in. 10 is a whole. Number Then we have this symbol here which represents the word and, so we can write and. And next we have Q, and the Q is the statement 5 is less than 3, so we can write that in, 5 is less. Tha 3 So this is the part inside the parentheses, but we also need to deal with the negation symbol outside of the parentheses. Now, one way that we can negate a statement is strategically adding in the word not, but to negate a conjunction, it can be tricky to determine where the word not would fit. So we can actually use another trick we've seen for finding negations. You can always create the negation by adding the phrase, it is not true that in front of a statement. To get a statement of the opposite meaning, AKA the negation. So that's what we're going to do here because it is a bit simpler. We're going to say. It is not true that to represent the negation, and then we have the conjunction after it, 10 is a whole number and 5 is less than 3. Now that we've translated it into words, we have to determine the truth value, and determining the truth value of these types of statements with parentheses can be really tricky. But let's think about it this way. Now, we want the negation of the conjunction P and Q, and we know that negations always have the opposite truth value to the original statement. So, to make it easier on ourselves, we can find the truth value of P and Q. Then we take the opposite truth value for the negation. So, let's look at just the conjunction part here for a second. Now the conjunction will be true if both P and Q are true individually, and the conjunction will be false if either P or Q is not true. Now, our first part, P, 10 is a whole number, is true, because 10 is a whole number. But our second part, 5 is less than 3, is false, because 5 is not less than 3. So if we're just looking at the conjunction P and Q, the conjunction is false. Right, because we don't have that both P and Q are true individually, so P and Q, the conjunction, is false. That means that the negation of the conjunction is true, because the negation of the conjunction has the opposite truth value. So because P and Q is false, we can say that not P and Q. Is true. Great work