Introduction to Mathematical Thinking. Tutorial for Assignment
1. Prove or disprove the statement “ All birds can fly. “ False. Counterexample.Ostrich Starting with the premise that all birds can fly, we will prove or disprove this statement by finding a counter-example. An obvious one is the ostrich, which cannot fly. 2. Prove or disprove the claim (forall x, y in R) [(x-y) ^2 >O] False. Counterexample. x=y=1. (x-y)^2=O To prove that all x in r, for all x and y in r, x minus y squared is greater than 0 is false,one must find a counter example. A counter example could be any pair of equal numbers. We will give a specific counter example by taking x equals y equals 1. In this case, x minus y squared equals 0 and 0 is not strictly based on 0 (it came close).If we excluded the cases when x or y was 0 then we would have a positive result, but it says that it’s true for all values of r not just one value, so any pair of equal numbers gives us a counterexample. 3. Prove that between any two unequal rationals there is a third rational. Let x , y in Q, x =y. Then x=p/q , y=r/s where p,q,r,s in Z Then x+y/2 = p/q+r/s/2 = ps+qr/2 = ps+qr/2qs in Q Rut x < x+y/2 < y. The third way is to prove that between any two unequal rational numbers there is athird. Let x and y be unequal rational numbers, with x less than y. Then because they are rational numbers, x is p/q and y is r/s where p, q, r, and s are integers. We must show that there is a number between them; let’s call this number x plus y over 2. If this number is a rational number then we have shown our result; but here is a proof that it is indeed rational: x plus y over 2 = p/q + r/s over qs/2 = ps + qr over 2 = (ps +qr)/2 = rational since it is the quotient of 2 integers. 7. Prove that sqrt3 is irrational. Prove it by contradiction. Assume sqrt3 were rational. Then sqrt3 = p/q , where p,q in N, with no common factors. Then 3=p^2/q^2 So 3q^2=p^ So 3|p^2 But 3 is prime So 3/p So p=3r So p^2=9r^2 So 3q^2=p^2=9r^2 So q^2=3r^2 So 3/q^2 So 3/q Contradiction, since p,q have no factors. ct proov fn sqrt2 p is even =2/p We will prove it by contradiction. We assume the square root of 3 is rational; that is, it can be expressed as a fraction p/q where p and q are natural numbers with no common factors. If that were true, then we would have 3 = p2/q2. Canceling out any common factors on both sides gives us 3 = p2. Let us multiply 3 times q squared. We will get 3q squared, which means p squared equals 3q squared. Since 3 is prime,and since a prime number divides any product of two numbers, the prime number 3 must divide one of these numbers. This means p is divisible by 3, or in other words,it can be written as 3r. So, if p is any number, then p squared equals nine times rsquared. We can take p squared and substitute it back in here to get 3q squared equals p squared, equals nine times r squared. So if we forget the middle term now,3q squared equals 9r squared. So if we divide both sides of this equation by 3, we’ll get q squared equals three times r squared, so that means three divides q. Now we have an inconsistency since p and q have no common factors. Yet we just showed that 3 is a common factor. So there is a contradiction. Just as with the proof of square root of 2, if we consider that p is even, then that’s just another way of saying 2 divides p. The proof that we have just seen for, the one in terms of even and odd,could be restated in terms of divisibility by three. The fact that 2 is prime is used in this proof as well. The only difference between this proof and the previous one is that we only use the fact that 2 is prime instead of 3. 1. Yes, suppose r+3 were rational. Then r + 3 = p/q where p, q, in Z. Then r = p/q -3 = p-3q/q in Q Contraditional Other similar. 8. Write down the converses of the following conditional statements: (a) If Dollar falls, the Yuan will rise If the Yuan rises, the Dollar falls. (b) If x<y then -y<-x (For x,y real numbers) ← T If -y<-x, then x<y ← T (c) If two triangles are congruent they have the same area. ← T If two triangles have the same area, then they are arguments. ← T The converse of a conditional statement is true only when the antecedent is false. If we swap the antecedent and consequent in a conditional statement and change the truth value of that statement, then we get its converse. For example, if the Yuanrises, then the dollar falls. The opposite case—if the dollar falls then the Yuanrises—is also true, reflecting the fact that these are two sides to the same coin. In this example, if x is less than y, then y must be greater than x; likewise if two triangles have equal areas, then they are congruent to each other. This exercise is intended to contrast it with the contrapositive. Also, we want to observe that truth and falsity can change. In this case,we start out with something that’s true – two triangles are congruent if and only if they have the same area. The converse of a true implication is also true: If two triangles are congruent, then they have the same area.It is true that if two triangles have the same area, they are congruent. It is false that if two triangles have different areas, they are not congruent. Sometimes the converse of a true statement can be false and sometimes it can be true. The situation with the contrapositive is different because when you swap around the order you can sometimes get through going to false and you can sometimes reserve truth. 11. Let r, s be irrationals. For each of the following, say whether the givennumber is necessarily irrational, and prove your answer. 1. r +3 2. 5r 3. r+3 4. rs 5. sqrt r 6. r^2 That was the topic of irrational numbers in the lecture. We saw that some of these are rational. We saw that this one can be rational, and there is an issue about this one being rational. Those were easy, so let’s move on to the more complicated stuff.If we assume that r plus 3 is rational, it can be written as p over q, where p and q are integers. However, if r equals p over q minus 3, then r must be rational and this contradicts the assumption that r is irrational. In each of these examples, we assume continuity by expressing it in terms of p over q, manipulate a bit and end up showing that r is rational in the square root example. 12. Let m and n be integers. Prove that: (a) If m and n are even, then m+n is even. (b) If m and n are even, then mn is divisible by 4. (с) If m and n are odd, then m+n is even. (d) If one of m,n is even and the other is odd, then m+n is odd. (e) If one of m, n is even and the other is odd, then mn is even. Key facts: n even iff n=2k, some k n add iff n=2k+1, some k (a) m=2k, n=2l, m+n=2k+2l=2(k+l) (d) m=2k, n=2l+1, m+n=2k+(2l+1)=2(k+l)+1 etc. The key facts to keep in mind when dealing with the even and odd numbers are that if n is even, then n is a multiple of 2 (whereas if n is odd, then n equals 2k + 1 for some integer k). The even numbers are therefore the multiples of 2, whereas the odd numbers are those that are 1 more than a multiple of 2. Between multiples of two, we can see that m is even if m is twice k and n is twice l. So for example, if m equals 2k, then n will be 2l. (Or vice versa.) This means that m plus n equals 2(2k) + 2(2l). If you put this into a calculator and press the equal sign, you’ll find that it’s 4 times something else. If one of the numbers is even, then it’s twice that number plus 1. The other number is odd: it’s equal to 2 times the first number plus 1. That works for all of them. We can do a tiny bit of algebra at this point, if we want to. The rest of this assignment is fairly straightforward; however, if you have not encountered proofs like this before in your studies, then they will likely be challenging.
