MATH220 April 2005
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Question 02

Consider the following two statements:
 $\mathbf {(a)} {\text{ for all }}w\in \mathbb {R} ,{\text{ there exists }}x\in \mathbb {R} {\text{ such that }}w<x.$
 $\mathbf {(b)} {\text{ there exists }}y\in \mathbb {R} {\text{ such that for all }}z\in \mathbb {R} {\text{, }}y<z.$
One of the statement is true, and the other is false. Determine which is which. For the true one, prove that it is true; for the false one, prove that it is false.

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

Try reformulating these two statements into more readable English.

Hint 2

The first statement is equivalent to
 Given any real number, there exists a real number larger than that one.

Hint 3

The second statement is equivalent to
 There exists a real number y that is strictly smaller than any real number.

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

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The second statement should appear wrong fairly immediately since it talking about the existence of the smallest real number which cannot exist. Let's prove this:
 The statement says that for any real number z it should be true that y < z, but if I pick z to be exactly that mysterious number y I should have y < y which is clearly impossible, so the statement is false.
Now we can prove that the first statement is true:
 That statement says that if any real number w is picked, then one can find a real number x that is larger than w. Well, sure enough, if one always choses x = w + 1, then clearly it is always going to be larger than the w that was picked in the first place.

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