Introduction to Mathematical Thinking. Tutorial for Assignment 2
Let us examine our data. One has been done for you. Two, the simple way to write that isto say 7 less than or equal to p less than 12. The next one we would write as 5 less than x less than 7. The next one, well if x is lessthan 4, then it's automatically less than 6. So the second conjunct here is superfluous; wecould just write that as x less than 4. What about the next one? Well, the second conjunct tells us that y is less than 3. But if ysatisfies this condition, then it must be less than 4, so the first conjunct is superfluous. Theonly one that counts is the second one. So we could simply write that as y squared lessthan 9. Consider the inequality x≤0. Since x≥0, it follows that 0≤x. However, since x≤0, it followsthat 0<x. Thus, there is only one possibility: x=0. To demonstrate that the conjunction is true, one would show that all of phi 1, phi 2, etc., upto phi n are true. Fourth, to prove that a conjunction is false is to show that at least one of its components isfalse. And so, you must find one or more of the components to be false—which wouldmean that the conjunction itself is also false. Okay, well that’s that one. If pi is greater than 10, then it’s automatically greater than 3.Therefore in terms of the disjunction, pi greater than 3 dominates—that is, it says more. Ifwe draw a picture, we get the following: 0, 3, 10. The first disjunct states that x is to the right of this point, and the second disjunct statesthat x is to the right of that point. The disjunction is true if at least one of these statementsis true. At least one statement will be true if we start at 3. First of all, only the first disjunct is correct. When we get beyond this point, both disjunctsare correct. So it’s the first one that works: that x is less than 0 or x is greater than 0. Thus the simplest way to write that is to say that x is not equal to 0. Thus if 0 is negative,or greater than 0, then x is positive. The standard abbreviation for greater than or equal to0 is x ≥ 0. Once you have x greater than or equal to 0, that's going to dominate the first disjunct interms of a disjunction. So you're going to have x greater than or equal to 0; that makes amore general claim of the two. And one might make a note that x squared greater than 9 means either x is greater than 3or x is less than negative 3. So, the statement then becomes simply: x squared greaterthan 9. That's the one that counts. As with numbers one and two, when we went through these options we articulated whatthey were. These options were just to prompt you to think about how you will express themin English. Okay, well that's that one. In number seven, you have to show that if disjunction is true,then at least one of the disjuncts must be true. And for the sake of argument, consider the following: If you wish to prove that statementnumber eight is false, you must demonstrate that all of phi 1 through phi n are false. Okay,well that takes care of numbers seven and eight. To say that pi is not greater than 3.2 is to say that pi is less than or equal to 3.2.
When you negate a strict greater than statement, you get less than or equal to. To say thatit's not the case that x is negative is to say that x is greater than or equal to 0. These arereal numbers because they come from the context of the expression, which is talkingabout real numbers. Every real number has a nonnegative square root, with the exception of 0. Thus, the onlyreal number for which it is not the case that x2 ≥ 0 is 0. The standard symbol for the phrase not x equals 1 is x not equal to 1. When you negate anegation, you get back to the original statement. This way we essentially answerednumber 10. We're returning to question 11 now. The answers I received are "Dollar strong and Yuanstrong" and "with this word, despite, but, in our mind." Those are just nuanced forms ofconjunction. We can say that despite the fact that the Yuan is weakening, and there is a tradeagreement, the dollar is strong. So despite and but are used to indicate whether somethingis contrary to expectations or consistent with expectations. They still say that all three of these things hold together. This one seems fairlystraightforward.
The dollar and the yuan cannot both be strong at the same time. If that were the case, itwould mean that both currencies would fail simultaneously. The trade agreement signed, but the dollar and yuan continue to fall. This indicates thatalthough the trade agreement was signed, it does not prevent a fall in the dollar or yuanvalues. And then the trade agreement between the United States and China failed, but bothcurrencies remained strong – the dollar strong and the yuan strong.