“Introduction to Mathematical Thinking. Tutorial for Assignment
By a denial of a statement ϕ we mean any statement equivalent to ⇁ϕ. Give a useful (and hence natural sounding) denial of each of the following statements. (a) 34,159 is a prime number. (b) Roses are red and violets are blue. (c) If there are no hamburgers, I’ll have a hot-dog. (d) Fred will go but he will not play. (e) The number x is either negative or greater than 10. (f) We will win the first game or the second. (a) The first one is a true statement: number 34,159 is prime (it has only two factors: 1and 34,159), but this is not what the question is about. The question asks for the negation or denial of the statement that 34,159 is prime. The simplest way to express this would beto say that 34,159 is not a prime number. If you are having trouble writing your answer in precise English, try using these other ways to express it: 34,159 is not an integer divisible by only two integers; or 34,159 is a composite number (it has more than two factors). (b) You would have to say that roses are not red or that violets are not blue, a sentence of the kind you would only see or hear in logic class as we study now. Mathematical uses of language require precision in expression, so it’s correct but not elegant. It can be tricky because there is a negation floating around in here. Remember when yougot a conditional phi-yield psi, when you deny it what you get is the antecedent conjoinedwith the negation of the consequence. (c) In this case, the antecedent is that there are no hamburgers. We need to be careful with formulations, so pay attention. The antecedent is there are no hamburgers. Conjoined with the fact that I won’t have a hot dog, I think it’s more natural to write the conjunction using but rather than and. And say, “but I won’t have a hot dog.” Okay, so we have an antecedent that there are no hamburgers conjoined with a but because it sounds more natural to say that as a but. But I won’t have a hot dog. Okay, well think about that one for a bit. (d) Students often have trouble with this sentence because of that negation, and it throws them off. So, you have to be a little bit careful. Fred will go or he’ll play, but the but is really the same as an “and,” so when we negate that, we get, “Fred won’t go” or “he’ll play.” Maybe in English we are more likely to write the sentence Fred will play, rather than Hewon’t go, because we would read into this some causality, or connection between the two. Now, of course, we would typically think of that as going beyond just the normal negation,but I think that’s how we understand it. So to me, it seems a more natural way of saying it.Just a clause with the kind of thing people usually do. But when it comes to negation, we need to think about the negation of each part and then the negation of “but”. This is because “or” is a disjunction in logic that means either A or B.So, I get “Fred will play” or “he won’t go”. This one is hard to translate into English but symbolically it’s very easy. The original statement is that x is negative or x is greater than ten. (e) If we negate the statement, the inequality x greater than or equal to 0 becomes an inequality of x less than 0. The or becomes and, x greater than 10 becomes x less than or equal to 10, so that just becomes 0, less than or equal to x, less than or equal to 10. If you do it in English you’re going to have to say something like “The number x is non-negative;you can’t say positive because negation of negative is not positive. Because of the way zero works, you must say a non-negative number, and the or becomes an and. You can’t say greater than ten if you mean not less than or equal to ten.You have to say less than or equal to 10. And I don’t see any way of doing it in English,other than with an awkward sounding sentence like this. (f) Symbols, such as the negation symbol, are more efficient than words in expressing ideas and solving logical problems. One answer to this question is: We will win the first game or lose the second. The negation of that statement is: We will lose both games. Towrite this in symbols we say: We lose the first two games. Remember the question asked for a denial, the answer to which was an equivalent version of the negation. And for some of these we were able to find pretty good answers. In other sit was kind of messy, but there were a couple of tricky ones floating around in there. 6. By a denial of a statement Ф we mean any statement equivalent to negФ. Give a useful (and hence natural sounding) denial of each of the following statements. (a) 34,159 is a prime number. 34,159 is not a prime number. (b) Roses are red and violets are blue. Roses are not red or violets are not blue. (c) If there are no hamburgers, I’ll have a hot-dog. neg[Ф => ψ]: Ф ^ negψ There are no hamburgers, but I won’t have a hot-dog. (d) Fred will go but he will not play. Fred won’t go or he’ll play. Fred will play or he won’t go. (e) The number x is either negative or greater than 10. The number x is non-negative and less or equal to 10. neg[(x < 0) v (x > 10)] (x≥0)^(x≤10) 0 ≤ x ≤ 10. (f) We will win the first game or the second. We’ll lose the first two games. So I would advise you to be careful about these kinds of things. Dealing with language precisely is not easy, as our minds tend to jump ahead from one idea to another. Howmany of you jumped from this statement to the idea of “If there are hamburgers, I won’thave a hotdog”? Nowadays someone often does that one, but they have to be careful about the precision of language. Human beings are very good at using everyday language, but the price is thatwe drop precision and think in terms of meanings rather than the literal meanings. The point of analyzing language is to be as precise as possible. Examples like this remindus that precision is not easy; we must work at it. Now to number 10. Because question nine is simply a truth table, we’ll refer you back to it for the answer. By now you should be able to knock off truth table arguments easily. Let us focus on the first of these statements, which is b. We need to show that this implies that, as well as that this implies that. We will proceed from left to right. This tells us that we can deduce psi and theta from phi, but we know that we can deduce psi and theta from psi and theta. Now, it’s just a conjunction, and if we know a conjunction,we know both of the conjuncts [or parts]. It follows that we can deduce psi and theta from phi. We can start with phi, deduce psi and theta, and then deduce those two formulas. And therefore, by chaining these things together, it follows that we can deduce psi and theta from phi. That is, we know that we can deduce Psi from Theta, and we know that we can deduce Psi from Phi, and we know that we can deduce Theta from Phi. Let’s begin with phi, and we’ll derive psi. Let’s begin with phi, and we’ll derive theta. Thus if we can do those two things, then we have a conjunct. If that’s established and that’s established, then we’ve established phi yields psi. And phi is then equal to theta. Well, the right-to-left direction was easy. How about the other way? Assume that this thing is true and deduce what he says about the other guy. Well, if we can deduce a conjunction,we can deduce each conjunct. So we can deduce that phi is true and that theta is true. Hence, we can deduce psi from theta and phi. We can also deduce theta from phi.However, if we can deduce psi from phi, and we can deduce theta from phi, it means that we can deduce psi and theta from phi. And if we can do that, that means that phi yields psi and theta. Well it’s a little bit intricate. As a first step, you need to keep track of what has been proven. Second, you must remember to call phi-squared and psi-squared different names. And when you are working on the logic, you may occasionally refer to something incorrectly. Check if you can do part c. Be sure not to take the following explanation literally, as it is intended only as an example of the kind of reasoning you should use while attempting this problem. Throughout this course, the worst thing you could do is look at any one answer and try to change the symbols and numbers in order to come up with another set of numbers that might work. If that was enough for high school, it won’t work for university mathematics. You need to go back and understand the method. So first: understand the reasoning behind it. That’s already a challenge at first, so don’t worry if you can’t do that right away. Once you’ve understood that, then have a try at (c). 10. Verify the equivalences in (b) and (c) in the previous questions by means of a logical argument. (So in the case of (b), for example, you must show that assuming Ф and deducing ψ^theta is the same as both deducing ψ from Ф and theta from Ф.) (b) [=>] We can deduce ψ^theta from Ф. But we can deduce ψ and theta from ψ^theta. It follows that we can deduce ψ and theta from Ф. That is Ф => ψ and Ф => 0. Hence (Ф => ψ) ^ (Ф =>theta). [Leftarrow] We can deduce (Ф => ψ) and (Ф =>theta). Hence, we can deduce ψ from Ф and we can deduce theta from Ф. Hence, we can deduce ( ψ ^theta) from Ф. Hence, Ф =>( ψ ^ theta). (c) Now try this yourself. Let’s move on to question 12. It concerns the contrapositive of four statements. Write down the contrapositives of these statements. If two rectangles have the same area, they are congruent. So, the contrapositive would be: If two rectangles do not have the same area, then they are not congruent. The converse of this statement is also true: if a triangle has sides with lengths a, b and c, with c the largest side; then if a + b squared is not equal to c squared, then the triangle is not right angled. The converse of this statement is also true. If in a triangle with sides a, b and c, where c is the largest side; then if a + b squared is not equal to c squared, then the triangle is not right angled. If the number is not prime, then the product of 2 and the reciprocal of that number is not prime; or if you like, if a number is composite then its reciprocal is also composite. And if the value of the dollar does not rise, then the value of the yuan will not rise. To solve this problem, you can use the contrapositive of a statement. The contrapositive changes the order of things and puts negation in front of them, so actually writing them down was not what the challenge was. The point of this exercise was to give you an example to sort of help cement the fact that the contrapositive is actually logically equivalent. Because, is this one, two, yes, congruent rectangles have the same area. If rectangles do not have the same areas, they are not congruent. These are both true. If a triangle is right angled, the Pythagorean theorem applies. If a triangle has a side inside it, each one of these is clear that one is equivalent to the other. If these things happen to be true, then other things will happen to be true; so I think these four examples will illustrate the fact that contrapositives are logically equivalent to the original statements. Question 14 was very similar; however, it asked us to prove some contrapositive statements rather than simply recognize their validity. 12. Write down the contrapositives of the following statements: (a) If two rectangles are congruent, they have the same area. If two rectangles do not have the same area, they are not congruent. (b) If a triangle with sides a,b,c (c largest) is right-angled, then a2+ b2= c2. If in a triangle with sides a, b, c (c largest), a2+b+c, then the triangle is not right angled. (c) If 2n-1 is prime, then n is prime. If n is not prime, then 2n – 1 is not prime. (d) If the Yuan rises, the Dollar will fall. If the dollar does not fall, then the Yuan won’t rise. (negψ) => (neg Ф) is called the contrapositive of Ф => ψ. The logical equivalence of a conditional and its contrapositive means that one way to prove an implication it is to verify the contrapositive. This is a common form of proof in mathematics that we’ll encounter later. Therefore, let’s do question 14 as well. So, here, you flip the order around but in this case you don’t inject any negation sign—simply the implication in the opposite direction. If two triangles have the same area, then they are congruent. Now that we have shown this to be true in one example, we can state the converse as well: if two triangles are congruent, then they have the same area. The converse does not necessarily follow from the original statement. Because the original statement is true, but the converse is false. Now we know this to be the case; let us examine its proof. The converse of this statement is that if, in a triangle with sides a, b and c with c the largest, it’s true that a2 + b2 = c2. Then the triangle is right angled. The converse of this statement is also true. So these statements are both true but what’s going on here is that what we’ve actually started out with is not just an implication when it’s actually an equivalent. When you have an equivalence, and take its converse, you get another statement that is equivalent to the original. The two statements have the same truth values when they are equivalent statements. In the case of this one, if n is prime and 2 to the n-1 is prime,then the converse would be that if the dollar falls, then the yuan will rise. And in this case there’s no reason to assume that one necessarily follows from the other, unless some way the other two currencies are linked. Okay? By the way, I hope you were able to write down the four converters correctly. But that wasn’t really the point of this exercise. The point was to show you that converses don’t necessarily yield equivalent statements. Whereas with contrapositives, we saw in example 12 that they do always yield equivalent statements. Okay, so I think this should take care of assignment #4. It is important not to confuse the contrapositive of a conditional Ф => ψ with its converse ψ => Ф. Use truth tables to show that the contrapositive and the converse of Ф => ψ are not equivalent. 14. Write down the converses of the four statements in question 12. (a) If two rectangles are congruent, they have the same area. T If two triangles have the same area, then they are congruent. F (b) If a triangle with sides a,b,c (e largest) is right-angled, then a2+b2=c2. T If in a triangle with sides a, b, c (c largest), a2+ b2= c2, then the triangle is right angled. T (c) If 2n -1 is prime, then n is prime. If n is prime, and 2n -1 is prime. (d) If the Yuan rises, the Dollar will fall. If the Dollar falls, the Yuan will rise.
