Science:Math Exam Resources/Courses/MATH220/April 2011/Question 08

MATH220 April 2011

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Question 08

Let $P\subset \mathbb {N}$ be the set of prime numbers $P=\{2,3,5,7,11,\ldots \}.$ Determine whether the following statements are true or false. Prove your answers ("true" or "false" is not sufficient). (a) $\displaystyle \forall m\in P,\forall n\in P,m+n\in P.$ (b) $\displaystyle \forall m\in P,\exists n\in P{\text{ s.t. }}m+n\in P.$ (c) $\displaystyle \exists m\in P{\text{ s.t. }}\forall n\in P,m+n\in P.$ (d) $\displaystyle \exists m\in P{\text{ s.t. }}\exists n\in P{\text{ s.t. }}m+n\in P.$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Try to translate those symbolic expressions into meaningful English for a start.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. For (a): First we can rewrite this statement without the symbols in slightly more meaningful English. For any prime number m and for any prime number n, m+n is prime. This is clear false since adding two primes doesn't always give a prime, for example 3+5=8. For (b): we get for any prime m there exists a prime n such that m+n is prime. For this to be true, it has to be true for any choice of prime m. Consider for example m=7. Can we add a prime to that number so that the sum is prime as well? There are two types of primes: the even ones (actually, 2 is the only even prime) and the odd ones. If we add 2 to 7, we get 9 which isn't prime, so that doesn't work. If we add any odd number to 7, we get an even number (clearly not 2) so clearly not a prime either. So no prime can be added to 7 to make it prime and hence the statement (b) is false. For (c): we get There exists a prime m such that for any prime n we have m+n is prime as well. This is fairly similar to the statement (b) except that here we want to be able to always add the same prime to any prime to get a new one. In other words, it is even harder to make this statement true and hence it is false. Indeed, we showed above that no prime can be added to 7 to make it prime again, so here 7 is a case of a value of n for which we show that no prime m would do the trick. For (d): we get There exists a prime m and there exists a prime n such that m+n is prime. This sounds much more reasonable and it is. This statement is true since we can actually even show values of m and n that work. Consider for example m=2 and n = 3. Then m+n=5 which is prime. So such values exist and so statement (d) is true. Note: actually in (d), you should easily convince yourself that one of the prime has to be 2 and so the other has to be what is called a twin prime, that is, a prime with the property that if you add 2 to it it is prime as well (those are, except for 2 and 3, the closest primes you might get and that's why we call them twin primes). For example, 5 and 7 are twin primes and so are 11 and 13; 17 and 19; 29 and 31. We do not know if there are infinitely many twin primes but we suppose it is the case (this is what we call the The twin prime conjecture).

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