Science:Math Exam Resources/Courses/MATH220/December 2009/Question 06 (b)

MATH220 December 2009

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Question 06 (b)

Let $\mathbb {I}$ be the set of irrational numbers. That is $\mathbb {I} =\mathbb {R} -\mathbb {Q}$. Decide whether the following statements are true or false. Prove your answers. Hint:The results from part (a) will be of great help. (i) $\forall {}x\in \mathbb {I} ,\forall {}y\in \mathbb {I} ,xy\in \mathbb {I}$. (ii) $\forall {}x\in \mathbb {I} ,\exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}xy\in \mathbb {I}$.

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
Part (i) claims that the product of any two irrational numbers is irrational. Part (i) claims that the product of any irrational number and some other irrational number is irrational.

Hint 2
For part (i) look at part (a) (i) and for part (ii) see if you can use both parts form part (a) together.

Hint 3
The first is false and the second is true.

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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. (i) This is false. For any irrational number x, choose $\displaystyle y={\frac {1}{x}}$ which is irrational by part (a)(i) of this problem. Then the product is 1 which is not irrational. This gives a counter example to the claim that the product of any two irrational numbers must be irrational. (ii) This is true. For any irrational number x, choose $\displaystyle y=1+{\frac {1}{x}}$ which is irrational by part (a)(i) and part (a)(ii) of this problem. Then the product is $\displaystyle x+1$ which is irrational again by part (a)(ii). Thus the claim is true that for any irrational number, there exists an irrational number such that their product is irrational.

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