Tutorial for Assignment 3

Introduction to Mathematical Thinking. Tutorial for Assignment New trade agreement will lead to strong currencies in both countries. So the trade agreement will result in telling us that both currencies are strong. That one is direct and easy to understand. The strong dollar indicates that the yuan is weak. It seems straightforward. The value of the dollar and the yuan are inversely correlated, so a weak dollar implies a weak yuan. The trade agreement failed because the dollar weakened, which was caused by news about the trade agreement failing. There cannot be another answer choice that would fit Part C as well. Part D is signaling very loudly that a trade agreement is signed and has implications for the economy. If a trade agreement is signed, then the strength of the U.S. dollar and Chinese yuan will be affected, so it's not accurate to say that one currency is strong while the other is weak. Now, with Part E, we're into an example where your mileage may. The question really is, what is a "but," and maybe the "but" means "and." Following, however, probably there could be two possible outcomes for the dollar. It could mean that the dollar is weak and the yuan is strong, or there is a trade agreement between China and the United States. The dollar is weak and the yuan is strong, which means the dollar is losing value and the yuan is gaining value. In other words, it's all conjunction, so you could interpret following it meaning that that happens first, and then those two happen. However, one could also argue that a new trade agreement led to these coincident strengths and weaknesses. One could argue that the dollar's weakness can be attributed to the fact that the Yuan is strong due to a new trade agreement. About Number 5, the situation is that the trade agreement was signed and a rise in the Yuan will result in a fall in the dollar. You could actually write a different way. You could say that there are two assumptions here. First, if a trade agreement is signed and second, if the Yuan rises, it follows that the Dollar falls. But these are equivalent as you could demonstrate using truth tables. There are two ways to interpret that. On the one hand, you could consider that you have an implication of an implication, or on the other hand, you could say that it's basically an assumption based on what these things mean. This is more of a literal interpretation and this one is more of a semantic interpretation where you look at what these things mean. If a trade agreement is signed, then these are linked. Equivalent. If you want to spread that out, you would say that a trade agreement implies that the dollar's rise leads to a yuan rise and that the yuan's rise leads to a dollar's rise. They are both the same thing. Vice versa. The bar conditional is an abbreviation for the two conditionals. Here, we are discussing an exclusive OR. So either the dollar is strong or the yen is strong. But since they are not both strong, we must conclude that one of them is strong and the other is weak. The dollar's strength is either due to its own value or the yen's weakness. But it can't be both; this forces an exclusive-or relationship. Both of them cannot hold, only one. We take newspaper headlines, which are abbreviated natural language, and try to interpret them in the formal language of logic. We're taking something that depends on context and knowledge of culture, and we're trying to express it in a logical formalism. That's a lot of interpretation differences. For number 2, I have put my answers in bold. If that doesn't show up on your screen, you will find these other choices involved. I have started with fi and negated it so that true becomes false, true becomes false, false becomes true, false becomes true. We have already worked out the truth table for the conditional and seen that it is T F T T. Now we can take not phi, or psi, and combine these two columns with a disjunction to get this guy. Whenever both A and B are true, we get a true value. So this will be true whenever A is true, which means that one of them's true here, neither is true here so we get a false. They're both true here, we get a true. One of them's true here we'll get a true. Combining these two conditions with this rule eliminates the need for a separate step to compare these two cases. By observing that every entry is the same, we can conclude that Phi yields Psi is equivalent to not Phi or Psi. This handles 2 and 3. We will now turn our attention to column 4. We have four columns to fill in. We begin with the two columns we're given. We negate the psi value in order to produce not-psi. So, a T became an F, an F became a T, T to F, F to T. Then we write down the values for phi yield psi, which we know. We've already worked that out, which T to F T to F T to F. Then, I negated those values to give me phi does not yield psi. T to F, F to T. We combined the first column with the third column by using conjunctions to give this one. With conjunctions, both parts must be present for something to be true. We got a T and then an F. The T means we're going to have an F there, so I get an F. Here, I have two Ts, so I get a T there. Here, I have one F, so I get an F there as well. Well, that closed on all of the columns. And then to answer question three, we just noticed that these things are all the same. And so, we see that phi does not yield psi. This fact can be expressed as a truth table by looking at phi and not psi as well as the negation of this operation, which is equivalent to phi yields psi. To do this, we must recall the discussion which led to the truth table for phi yields psi. If you remember, the discussion went as follows. We had to look at the statement "phi does not yield psi." To discover the truth values in a case where phi is false, we had to look at phi does not yield psi. This truth table shows why this enabled us to find the problematic truth values for yields. For implication, when phi is true if and only if psi is false. Because it is this equivalence that we were capitalizing on. That phi does not yield psi if phi is true, and anyway, psi is false. So, that's questions four and five for you. How is your progress? A puzzle for the ending. And that would just give you one clue. The focus is on precision in language and on conveying information accurately. With that clue, you are left to puzzle out this problem on your own.