Science:Math Exam Resources/Courses/MATH220/December 2010/Question 07 (a)

MATH220 December 2010

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Question 07 (a)

For each subset of $\mathbb {R}$ give its supremum, maximum, infimum, and minimum if they exist. If they do not exist write "none". You do not need to justify your answers. the sets sup max inf min $\{x\in \mathbb {R} \quad {\textrm {s.t.}}\quad {}2\leq {x}<\pi \}$ $\{x\in \mathbb {Q} \quad {\textrm {s.t.}}\quad {}x\geq 0\quad {\textrm {and}}\quad {}x^{2}\leq 5\}$ $\{x\in \mathbb {Z} \quad {\textrm {s.t.}}\quad {}x^{2}\geq 4\}$ $\bigcup _{n=1}^{\infty }\left[{\frac {1}{n}},n^{2}+1\right]$

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
Remember that max and mins (if they exist) are contained inside the set whereas supremums and infimums are always real numbers (in this course) and may or may not be inside the set.

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. $\{x\in \mathbb {R} \quad {\textrm {s.t.}}\quad {}2\leq {x}<\pi \}$ Supremum: $\displaystyle \pi$. All elements in this set are less than $\displaystyle \pi$ and this is the least upper bound of this set as anything slightly smaller than $\pi$ is inside the set. Maximum: None. The supremum is not in the set and thus it has no maximum. Infimum: $\displaystyle 2$. All elements in this set are at least as large as (or equal to) 2 and this is the greatest lower bound as anything less than 2 is strictly not in the set. Minimum: $\displaystyle 2$. The infimum is inside the set and hence it is also the minimum. $\{x\in \mathbb {Q} \quad {\textrm {s.t.}}\quad {}x\geq 0\quad {\textrm {and}}\quad {}x^{2}\leq 5\}$ Supremum: $\displaystyle {\sqrt {5}}$. This element is the largest real number satisfying $x^{2}\leq 5$ and any real number larger than this element has a square larger than 5. Maximum: None. Since the supremum is irrational, it is not contained in this set and so this set has no maximum. Infimum: $\displaystyle 0$. Since all the elements of this set are nonnegative and this set contains $0$, this must be the infimum. Minimum: $\displaystyle 0$. The set contains the infimum and hence it is the minimum. $\{x\in \mathbb {Z} \quad {\textrm {s.t.}}\quad {}x^{2}\geq 4\}$ Supremum: None. The set contains all positive numbers greater than or equal to $2$ and thus has no least upper bound. Maximum: None. Without a supremum, there can be no maximum. Infimum: None. The set contains all negative numbers less than or equal to $-2$ and thus has no greatest lower bound. Minimum: None. Without an infimum, there can be no minimum. $\bigcup _{n=1}^{\infty }\left[{\frac {1}{n}},n^{2}+1\right]$ Supremum: None. The right endpoints continually grow to infinity. If there were a supremum $c$, simply find an integer $n$ larger than $c$ and this will show that in fact $c$ is inside this infinite union. Maximum: None. Without a supremum, there can be no maximum. Infimum: $\displaystyle 0$. Since the left endpoints are shrinking to $0$ and we can get as close as we like to $0$, this is the infimum. Minimum: None. The infimum is not an element of the set.

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