Math Problems Vault

Problem I.6.1

Prove that for arbitrary cardinalities $m$ and $n$ exactly one of the conditions $m = n$ , $m < n$ , or $n < m$ holds (trichotomy).

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Problem I.6.2

Prove that the cardinal numbers are linearly ordered by the relation $\leq$ .

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Problem I.6.3

Prove that there is no greatest cardinal number.

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Problem I.6.4

Prove that:

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(a)

$3 + 5 = 8$ ;

(b)

$n + \aleph_{0} = \aleph_{0}$ , where $n$ is finite;

(c)

$\aleph_{0} + \aleph_{0} = \aleph_{0}$ ;

(d)

$\aleph_{0} + c = c$ ;

(e)

$c + c = c$ .

Problem I.6.5

(a)(b)

Prove that:

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(a)

for any sets $A_{1}, A_{2}$ there exist sets $B_{1}, B_{2}$ such that $A_{1} \sim B_{1}$ , $A_{2} \sim B_{2}$ , and $B_{1} \cap B_{2} = \emptyset$ ;

(b)

the sum of two cardinal numbers always exists.

Problem I.6.6

(a)(b)(c)

Prove for arbitrary cardinal numbers:

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(a)

$n_{1} + n_{2} = n_{2} + n_{1}$ ;

(b)

$n_{1} + (n_{2} + n_{3}) = (n_{1} + n_{2}) + n_{3}$ ;

(c)

$n + 0 = n$ .

Problem I.6.7

(a)(b)(c)(d)

Let $A$ , $B$ , $C$ , $A_{1}, \ldots , A_{n}$ be finite sets. Prove that:

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(a)

(b)

(c)

$\left|\bigcup_{i=1}^{n} A_{i}\right| = \sum_{i=1}^{n} \left|A_{i}\right| + \ldots + (-1)^{k-1} \sum_{\substack {i_{1}, \ldots , i_{k} = 1 \\ i_{1} < \ldots < i_{k}}}^{n} \left|A_{i_{1}} \cap \ldots \cap A_{i_{k}}\right| + \ldots ;$

(d)

$\left|A_{1} \setminus A_{2} \setminus \ldots \setminus A_{n}\right| = \sum_{i=1}^{n} \left|A_{i}\right| + \ldots + (-2)^{k-1} \sum_{\substack {i_{1}, \ldots , i_{k} = 1 \\ i_{1} < \ldots < i_{k}}}^{n} \left|A_{i_{1}} \cap \ldots \cap A_{i_{k}}\right| + \ldots$

Problem I.6.8

(a)(b)

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(a)

Prove that $n < m \Rightarrow n + 1 \leq m$ .

(b)

Give an example of cardinal numbers $n$ and $m$ such that $n + 1 \leq m$ , but $m \leq n$ .

Problem I.6.9

Prove that $n \leq m$ if and only if there exists $n_{1}$ such that $n + n_{1} = m$ . Show that such an $n_{1}$ is not uniquely determined.

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Problem I.6.10

(a)(b)(c)(d)

Prove that:

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(a)

$3 \cdot 5 = 15$ ;

(b)

$\aleph_{0} \cdot \aleph_{0} = \aleph_{0}$ ;

(c)

$\aleph_{0} \cdot c = c$ ;

(d)

$c \cdot c = c$ .

Problem I.6.11

(a)(b)

Prove that:

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(a)

if $A \sim B$ and $C \sim D$ , then $A \times C \sim B \times D$ ;

(b)

the product of two cardinal numbers always exists.

Problem I.6.12

Prove for arbitrary cardinal numbers:

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(a)

$n_{1} \cdot n_{2} = n_{2} \cdot n_{1}$ ;

(b)

$n_{1} \cdot (n_{2} \cdot n_{3}) = (n_{1} \cdot n_{2}) \cdot n_{3}$ ;

(c)

$n_{1} \cdot (n_{2} + n_{3}) = (n_{1} \cdot n_{2}) + (n_{1} \cdot n_{3})$ ;

(d)

$n \cdot 1 = n$ ;

(e)

$n \cdot 0 = 0$ ;

(f)

$n \cdot m = n$ , if $m$ is finite and $n$ is an infinite cardinal number;

(g)

$n \cdot \aleph_{0} = n$ , if $n$ is an infinite cardinal number.

Problem I.6.13

Prove that $n^{2} = n$ , if $n$ is an infinite cardinal number.

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Problem I.6.14

Prove that if $n$ and $m$ are cardinal numbers and one of them is infinite, then $n \cdot m = n + m = \max (n, m)$ , provided $n \neq 0$ and $m \neq 0$ .

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Problem I.6.15

(a)(b)(c)

Prove that:

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(a)

$2^{\aleph_{0}} = c$ ;

(b)

$\aleph_{0}^{\aleph_{0}} = c$ ;

(c)

$c^{\aleph_{0}} = c$ .

Problem I.6.16

Prove that for two cardinal numbers $n$ and $m$ , $n^{m}$ always exists.

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Problem I.6.17

Prove for arbitrary cardinal numbers $m$ , $n$ , and $p$ :

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(a)

$m^{n+p} = m^{n} \cdot m^{p}$ ;

(b)

$(m \cdot n)^{p} = m^{p} \cdot n^{p}$ ;

(c)

$(m^{n})^{p} = m^{n \cdot p}$ ;

(d)

$m^{1} = m$ ;

(e)

$1^{m} = 1$ .

Problem I.6.18

Prove that $\left|P(A)\right| = 2^{\left|A\right|}$ .

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Problem I.6.19

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Problem I.6.20

(a)(b)

Prove for arbitrary cardinal numbers:

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(a)

if $2^{m} \geq \aleph_{0}$ , then $2^{m} \geq 2^{\aleph_{0}}$ ;

(b)

if $m^{n} = \aleph_{0}$ , then $m = \aleph_{0}$ , and $n$ is finite.

Problem I.6.21

(a)(b)

Prove for arbitrary cardinal numbers:

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(a)

if $n \geq \aleph_{0}$ , then $2^{n} = n^{n}$ ;

(b)

if $1 < m \leq n$ , $\aleph_{0} \leq n$ , then $m^{n} = n^{n}$ .

Problem I.6.22

(a)(b)(c)

Prove for arbitrary cardinal numbers:

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(a)

if $\left|I\right| = m$ , $n_{i} = n$ for all $i \in I$ , then $m \cdot n = \sum_{i \in I} n_{i}$ ;

(b)

if $m + p = n + p$ and $p$ is finite, then $m = n$ ;

(c)

if $2 \cdot n_{1} = 2 \cdot n_{2}$ , then $n_{1} = n_{2}$ .

Problem I.6.23

Prove that

\left|\bigcup _{i \in I} A_{i}\right| \leq \sum _{i \in I} \left|A_{i}\right|.

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Problem I.6.24

(a)(b)(c)

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(a)

Let $\left\{ m_{i}\right\}$ $(i \in I)$ be a family of cardinal numbers, with $m_{i} = 0$ for $i \in J \subseteq I$ . Prove that

\sum _{i \in I} m_{i} = \sum _{i \in I \setminus J} m_{i}.

(b)

Let $\left\{ m_{i}\right\}$ $(i \in I)$ be a family of cardinal numbers, with $m_{i} = 1$ for $i \in J \subseteq I$ . Prove that

\prod _{i \in I} m_{i} = \prod _{i \in I \setminus J} m_{i}.

(c)

Prove that $\prod_{i \in I} m_{i} = 0$ if and only if there exists $i_{0} \in I$ such that $m_{i_{0}} = 0$ .

Problem I.6.25

(a)(b)

Prove that if $\varphi$ is a permutation of the set $I$ , then:

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(a)

$\sum_{i \in I} m_{i} = \sum_{i \in I} m_{\varphi (i)}$ ;

(b)

$\prod_{i \in I} m_{i} = \prod_{i \in I} m_{\varphi (i)}$ .

Problem I.6.26

(a)(b)

Let $\left\{ A_{\lambda }\right\}_{\lambda \in L}$ be a partition of the set $J$ into nonempty pairwise disjoint sets. Prove that:

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(a)

$\sum_{i \in I} m_{i} = \sum_{\lambda \in L} \left(\sum_{i \in A_{\lambda }} m_{i}\right)$ ;

(b)

$\prod_{i \in I} m_{i} = \prod_{\lambda \in L} \left(\prod_{i \in A_{\lambda }} m_{i}\right)$ .

Problem I.6.27

(a)(b)

Prove that:

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(a)

$n \cdot \sum_{i \in I} m_{i} = \sum_{i \in I} (n \cdot m_{i})$ ;

(b)

$\prod_{i \in I} \left(\sum_{j \in J_{i}} m_{ij}\right) = \sum_{f \in K} \prod_{i \in I} m_{i f(i)}$ , where $K = \prod_{i \in I} J_{i}$ .

Problem I.6.28

(a)(b)

Prove that if $m_{i} \leq n_{i}$ for all $i \in I$ , then:

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(a)

$\sum_{i \in I} m_{i} \leq \sum_{i \in I} n_{i}$ ;

(b)

$\prod_{i \in I} m_{i} \leq \prod_{i \in I} n_{i}$ .

Problem I.6.29

(a)(b)

Let $J \subseteq I$ . Prove that:

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(a)

$\sum_{i \in J} m_{i} \leq \sum_{i \in I} m_{i}$ ;

(b)

$\prod_{i \in J} m_{i} \leq \prod_{i \in I} m_{i}$ , where $m_{i} \neq 0$ for $i \in I \setminus J$ .

Problem I.6.30

(a)(b)

Let $\left\{ m_{i}\right\}_{i \in I}$ and $\left\{ n_{i}\right\}_{i \in I}$ be two families of cardinal numbers, with $n_{i} \geq 2$ for all $i \in I$ . Prove that:

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(a)

if $m_{i} \leq n_{i}$ for all $i \in I$ , then $\sum_{i \in I} m_{i} \leq \prod_{i \in I} n_{i}$ ;

(b)

if $m_{i} < n_{i}$ for all $i \in I$ , then $\sum_{i \in I} m_{i} < \prod_{i \in I} n_{i}$ .

Problem I.6.31

Let $0 < m_{0} < m_{1} < \ldots$ Prove that $\sum_{n \in \mathbb {N}} m_{n} < \prod_{n \in \mathbb {N}} m_{n}$ .

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Problem I.6.32

Let $n \geq \aleph_{0}$ , $\left\{ n_{i}\right\}_{i \in I}$ be a family of cardinal numbers not exceeding $n$ , $\left|I\right| \leq n$ . Prove that $\sum_{i \in I} n_{i} \leq n$ .

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Problem I.6.33

Let $\alpha$ and $\beta$ be cardinal numbers, $\left|I\right| = \beta$ , and $\alpha_{i} = \alpha$ for every $i \in I$ . Prove that $\alpha^{\beta } = \prod_{i \in I} \alpha_{i}$ .

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Problem I.6.34

(a)(b)

Prove that:

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(a)

$m^{\left(\sum_{i \in I} n_{i}\right)} = \prod_{i \in I} m^{n_{i}}$ ;

(b)

$\left(\prod_{i \in I} m_{i}\right)^{n} = \prod_{i \in I} m_{i}^{n}$ .

Problem I.6.35

Prove that for any cardinal number $m$ , $m^{\aleph_{0}}$ cannot be represented in the form $\sum_{i \in \mathbb {N}} m_{i}$ , where $m_{i} < m_{i+1}$ .

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