Science:Math Exam Resources/Courses/MATH220/December 2010/Question 03 (c)

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MATH220 December 2010

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Question 03 (c)

Let $\mathbb {I}$ be the set of irrational numbers. That is $\mathbb {I} =\mathbb {R} -\mathbb {Q}$. Determine whether the following statements are true or false --- explain your answers ("true" or "false" is not sufficient). (i) $\forall {}x\in \mathbb {I} ,\forall {}y\in \mathbb {I} ,x+y\in \mathbb {I}$ (ii) $\forall {}x\in \mathbb {I} ,\exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}x+y\in \mathbb {I}$ (iii) $\exists {}x\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}\forall {}y\in \mathbb {I} ,x+y\in \mathbb {I}$ (iv) $\exists {}x\in \mathbb {I} \quad {\textrm {s.t.}}\quad \exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}x+y\in \mathbb {I}$

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
The previous two parts should be really valuable here.

Hint 2
Try to rewrite each of these statements in English. For example, (i) $\forall {}x\in \mathbb {I} ,\forall {}y\in \mathbb {I} ,x+y\in \mathbb {I}$ For all irrational numbers x and y, we have that their sum must be irrational. Does this sentence sound correct? Can you use the previous two parts to help find counter examples?

Hint 3

For the other parts: (ii) $\forall {}x\in \mathbb {I} ,\exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}x+y\in \mathbb {I}$ For every irrational number x, there exists an irrational number y such that their sum is irrational. (iii) $\exists {}x\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}\forall {}y\in \mathbb {I} ,x+y\in \mathbb {I}$ There exists an irrational number such that for all irrational numbers y, their sum is always irrational. (iv) $\exists {}x\in \mathbb {I} \quad {\textrm {s.t.}}\quad \exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}x+y\in \mathbb {I}$ There exists irrational numbers x and y such that their sum is irrational.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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

We proceed statement by statement using parts (a) and (b) (i) $\forall {}x\in \mathbb {I} ,\forall {}y\in \mathbb {I} ,x+y\in \mathbb {I}$ This is false. For example, pick any irrational number (say $\displaystyle x={\sqrt {2}}$) and then notice that if $\displaystyle y=-{\sqrt {2}}$, then the sum is 0 which is a rational number. (ii) $\forall {}x\in \mathbb {I} ,\exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}x+y\in \mathbb {I}$ This is true. For any such x, just take y to be equal to x and by part b we see that $\displaystyle x+y=2x\in \mathbb {I}$ (iii) $\exists {}x\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}\forall {}y\in \mathbb {I} ,x+y\in \mathbb {I}$ This is false. No matter what value of x you choose, its negation is always irational and by the logic above for part (i), we see that this sum is 0, a rational number. (iv) $\exists {}x\in \mathbb {I} \quad {\textrm {s.t.}}\quad \exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}x+y\in \mathbb {I}$ This is true. Pick any irrational number (say $\displaystyle x={\sqrt {2}}$) and then let y equal x. Then use part (b) to show that $\displaystyle x+y=x+x=2x\in \mathbb {I}$.

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