Math Problems Vault

Problem 2A.1

Prove that if $A$ and $B$ are subsets of $\mathbb {R}$ and $\left|B\right| = 0$ , then $\left|A \cup B\right| = \left|A\right|$ .

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Problem 2A.2

Suppose $A \subseteq \mathbb {R}$ and $t \in \mathbb {R}$ . Let $t A = \left\{ t a: a \in A\right\}$ . Prove that $\left|t A\right| = \left|t\right|\left|A\right|$ .

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Remark.

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Assume that $0 \cdot \infty$ is defined to be 0.

Problem 2A.3

Prove that if $A, B \subseteq \mathbb {R}$ and $\left|A\right| < \infty$ , then $\left|B \backslash A\right| \geq \left|B\right|-\left|A\right|$ .

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Problem 2A.4

Suppose $F$ is a subset of $\mathbb {R}$ with the property that every open cover of $F$ has a finite subcover. Prove that $F$ is closed and bounded.

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Problem 2A.5

Suppose $\mathcal{A}$ is a set of closed subsets of $\mathbb {R}$ such that $\bigcap_{F \in \mathcal{A}} F = \varnothing$ . Prove that if $\mathcal{A}$ contains at least one bounded set, then there exist $n \in \mathbb {N}$ and $F_{1}, \ldots , F_{n} \in \mathcal{A}$ such that $F_{1} \cap \cdots \cap F_{n} = \varnothing$ .

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Problem 2A.6

Prove that if $a, b \in \mathbb {R}$ and $a < b$ , then

\left|(a, b)\right| = \left|[a, b)\right| = \left|(a, b]\right| = b-a.

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Problem 2A.7

Suppose $a, b, c, d$ are real numbers with $a < b$ and $c < d$ . Prove that

\left|(a, b) \cup (c, d)\right| = (b-a)+(d-c) \text{ if and only if }(a, b) \cap (c, d) = \varnothing

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Problem 2A.8

Prove that if $A \subseteq \mathbb {R}$ and $t > 0$ , then $\left|A\right| = \left|A \cap (-t, t)\right|+\left|A \cap (\mathbb {R} \backslash (-t, t))\right|$ .

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Problem 2A.9

Prove that $\left|A\right| = \lim_{t \rightarrow \infty }\left|A \cap (-t, t)\right|$ for all $A \subseteq \mathbb {R}$ .

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Problem 2A.10

Prove that $\left|[0,1] \backslash \mathbf{Q}\right| = 1$ .

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Problem 2A.11

Prove that if $I_{1}, I_{2}, \ldots$ is a disjoint sequence of open intervals, then

\left|\bigcup _{k = 1}^{\infty } I_{k}\right| = \sum _{k = 1}^{\infty } \ell \left(I_{k}\right)

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Problem 2A.12

(a)(b)(c)

Suppose $r_{1}, r_{2}, \ldots$ is a sequence that contains every rational number. Let

F = \mathbb {R} \backslash \bigcup _{k = 1}^{\infty }\left(r_{k}-\frac{1}{2^{k}}, r_{k}+\frac{1}{2^{k}}\right)

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(a)

Show that $F$ is a closed subset of $\mathbb {R}$ .

(b)

Prove that if $I$ is an interval contained in $F$ , then $I$ contains at most one element.

(c)

Prove that $\left|F\right| = \infty$ .

Problem 2A.13

Suppose $\varepsilon > 0$ . Prove that there exists a subset $F$ of $[0,1]$ such that $F$ is closed, every element of $F$ is an irrational number, and $\left|F\right| > 1-\varepsilon$ .

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Problem 2A.14

Consider the following figure, which is drawn accurately to scale.

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(a)

Show that the right triangle whose vertices are $(0,0),(20,0)$ , and $(20,9)$ has area 90.

(b)

Show that the yellow (lower) right triangle has area 27.5.

(c)

Show that the red rectangle has area 45.

(d)

Show that the blue (upper) right triangle has area 18.

(e)

Add the results of parts (b), (c), and (d), showing that the area of the colored region is 90.5.

(f)

Seeing the figure above, most people expect parts (a) and (e) to have the same result. Yet in part (a) we found area 90, and in part (e) we found area 90.5. Explain why these results differ.

Remark.

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Regarding (a): we have not defined area yet, but just use the elementary formulas for the areas of triangles and rectangles that you learned long ago.

Regarding (f): you may be tempted to think that what we have here is a two-dimensional example similar to the result about the nonadditivity of outer measure (2.18). However, genuine examples of nonadditivity require much more complicated sets than in this example.