Math Problems Vault

Problem I.1.1

Prove:

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(a)

$A \subseteq A$ (reflexivity);

(b)

if $A \subseteq B$ and $B \subseteq C$ , then $A \subseteq C$ (transitivity);

(c)

$A \cap B \subseteq A \subseteq A \cup B$ ;

(d)

$A \cap B \subseteq B \subseteq A \cup B$ ;

(e)

$A \setminus B \subseteq A$ .

Problem I.1.2

Prove that if $A$ is the set of roots of the equation $x^{2} - 7x + 6 = 0$ and $B = \left\{ 1, 6\right\}$ , then $A = B$ .

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Problem I.1.3

Prove that $\emptyset \neq \left\{ \emptyset \right\}$ .

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Problem I.1.4

Prove that $\left\{ \left\{ 1, 2\right\} , \left\{ 2, 3\right\} \right\} \neq \left\{ 1, 2, 3\right\}$ .

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Problem I.1.5

(a)(b)(c)

Prove that for any $A$ :

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(a)

$\emptyset \subseteq A \subseteq U$ ;

(b)

if $A \subseteq \emptyset$ , then $A = \emptyset$ ; if $U \subseteq A$ , then $A = U$ ;

(c)

$A \cup \emptyset = A$ , $A \cap \emptyset = \emptyset$ , $A \cup U = U$ , $A \cap U = A$ .

Problem I.1.6

Prove that there exists only one set with no elements.

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Problem I.1.7

Do there exist sets $A$ , $B$ , and $C$ such that

A \cap B \neq \emptyset , \quad A \cap C = \emptyset , \quad (A \cap B) \setminus C = \emptyset \text{?}

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Problem I.1.8

Prove that the set of all roots of the polynomial $\alpha (x) = \beta (x) \cdot \gamma (x)$ is the union of the sets of roots of the polynomials $\beta (x)$ and $\gamma (x)$ .

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Problem I.1.9

Prove that the intersection of the sets of real roots of the polynomials $\alpha (x)$ and $\beta (x)$ with real coefficients coincides with the set of all real roots of the polynomial $\gamma (x) = \alpha^{2}(x) + \beta^{2}(x)$ .

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Problem I.1.10

Prove that

A \subseteq B \Leftrightarrow A \cup B = B \Leftrightarrow A \cap B = A \Leftrightarrow A \setminus B = \emptyset \Leftrightarrow (-A) \cup B = U.

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Problem I.1.11

Prove the following identities:

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(a)

$A \cup A = A \cap A = A$ ;

(b)

$A \cap B = B \cap A$ ;

(c)

$A \cup B = B \cup A$ ;

(d)

$A \cap (B \cap C) = (A \cap B) \cap C$ ;

(e)

$A \cup (B \cup C) = (A \cup B) \cup C$ ;

(f)

$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ ;

(g)

$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ ;

(h)

$(A \cap B) \cup (C \cap D) = (A \cup C) \cap (B \cup C) \cap (A \cup D) \cap (B \cup D)$ .

Problem I.1.12

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

Problem I.1.13

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

Problem I.1.14

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Problem I.1.15

(a)(b)

Prove that:

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(a)

$(A_{1} \cup \ldots \cup A_{n}) \symdiff (B_{1} \cup \ldots \cup B_{n}) \subseteq (A_{1} \symdiff B_{1}) \cup \ldots \cup (A_{n} \symdiff B_{n})$ ;

(b)

$(A_{1} \cap \ldots \cap A_{n}) \symdiff (B_{1} \cap \ldots \cap B_{n}) \subseteq (A_{1} \symdiff B_{1}) \cup \ldots \cup (A_{n} \symdiff B_{n})$ .

Problem I.1.16

(a)(b)(c)

Prove that:

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(a)

$A \symdiff B = \emptyset \Leftrightarrow A = B$ ;

(b)

$A \cap B = \emptyset \Rightarrow A \cup B = A \symdiff B$ ;

(c)

$A \symdiff B = C \Leftrightarrow B \symdiff C = A \Leftrightarrow C \symdiff A = B$ .

Problem I.1.17

(a)(b)(c)

Define the operations $\cup , \cap , \setminus$ in terms of:

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(a)

$\symdiff , \cap$ ;

(b)

$\symdiff , \cup$ ;

(c)

$\setminus , \symdiff$ .

Problem I.1.18

(a)(b)

Prove that it is impossible to define:

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(a)

$\setminus$ in terms of $\cap$ and $\cup$ ;

(b)

$\cup$ in terms of $\cap$ and $\setminus$ .

Problem I.1.19

Prove that sets form a ring without identity, where $\symdiff$ plays the role of addition and $\cap$ plays the role of multiplication. What plays the role of subtraction in this ring?

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Problem I.1.20

Find all subsets of the sets $\emptyset$ , $\left\{ \emptyset \right\}$ , $\left\{ x\right\}$ , $\left\{ 1, 2\right\}$ .

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Problem I.1.21

(a)(b)

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(a)

Prove that a set of $n$ elements has $2^{n}$ subsets.

(b)

How many subsets of $k$ elements does a set of $n$ elements have $(k \leq n)$ ?

Problem I.1.22

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

Prove that:

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(a)

$P(A \cap B) = P(A) \cap P(B)$ ;

(b)

$P\left(\bigcap_{i \in I} A_{i}\right) = \bigcap_{i \in I} P(A_{i})$ ;

(c)

$P(A \cup B) = \left\{ A_{1} \cup B_{1} \mid A_{1} \in P(A) \text{ and } B_{1} \in P(B)\right\}$ ;

(d)

$P\left(\bigcup_{i \in I} A_{i}\right) = \bigcup_{i \in I} \left\{ B_{i} \mid B_{i} \in P(A_{i})\right\}$ .

Problem I.1.23

Prove that for any $a$ , $b$ , $c$ , $d$

\left\{ \left\{ a\right\} , \left\{ a, b\right\} \right\} = \left\{ \left\{ c\right\} , \left\{ c, d\right\} \right\} \Leftrightarrow a = c \text{ and } b = d.

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Problem I.1.24

Which of the following statements hold for all $A$ , $B$ , and $C$ :

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(a)

if $A \in B$ and $B \in C$ , then $A \in C$ ?

(b)

if $A \subseteq B$ and $B \in C$ , then $A \in C$ ?

(c)

if $A \cap B \subseteq -C$ and $A \cup C \subseteq B$ , then $A \cap C = \emptyset$ ?

(d)

if $A \neq B$ and $B \neq C$ , then $A \neq C$ ?

(e)

if $A \subseteq -(B \cup C)$ and $B \subseteq -(A \cup C)$ , then $B = \emptyset$ ?

Problem I.1.25

Prove that for any $A_{1}, A_{2}, \ldots , A_{n}$ ,

\text{if } A_{1} \subseteq A_{2} \subseteq \ldots \subseteq A_{n} \subseteq A_{1}, \text{ then } A_{1} = A_{2} = \ldots = A_{n}.

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Problem I.1.26

For every positive integer $n$ , exhibit a set $A_{n}$ of $n$ elements such that if $x, y \in A_{n}$ , then $x \in y$ or $y \in x$ or $x = y$ .

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Problem I.1.27

Solve the system of equations

\begin{cases} A \cap X = B, \\ A \cup X = C, \end{cases}

where $A$ , $B$ , and $C$ are given sets and $B \subseteq A \subseteq C$ .

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Problem I.1.28

Solve the system of equations

\begin{cases} A \setminus X = B, \\ X \setminus A = C, \end{cases}

where $A$ , $B$ , and $C$ are given sets and $B \subseteq A$ , $A \cap C = \emptyset$ .

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Problem I.1.29

(a)(b)

Let systems of sets $\left\{ A_{i}\right\}_{i \in I}$ and $\left\{ B_{i}\right\}_{i \in I}$ be given, where $I$ is some set. Solve the systems of equations:

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(a)

$A_{i} \cap X = B_{i}$ , $i \in I$ ;

(b)

$A_{i} \cup X = B_{i}$ , $i \in I$ .

For which $A_{i}$ and $B_{i}$ do these systems have solutions?

Problem I.1.30

Solve the system of equations

\begin{cases} A \setminus X = B, \\ A \cup X = C, \end{cases}

where $A$ , $B$ , and $C$ are given sets and $B \subseteq A \subseteq C$ .

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Problem I.1.31

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

Show that:

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(a)

$A = B \Leftrightarrow (A \setminus B) \cup (B \setminus A) = \emptyset$ ;

(b)

any equation in the set $X$ whose right-hand side is $\emptyset$ is equivalent to the equation $(A \cap X) \cup [B \cap (-X)] = \emptyset$ , where $A$ and $B$ are some sets whose expressions do not contain the symbol $X$ ;

(c)

the system of equations

\begin{cases} A \cap X = \emptyset , \\ B \cap (-X) = \emptyset \end{cases}

has a solution if and only if $B \subseteq -A$ ; under this condition, a solution of the system is any set $X$ such that

B \subseteq X \subseteq -A;

(d)

describe a method for solving a system of equations with one unknown.

Problem I.1.32

(a)(b)(c)

Using the method of problem I.1.31, solve the following systems:

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(a)

$\begin{cases} A \cup X = B \cap X, \\ A \cap X = C \cup X; \end{cases}$

(b)

$\begin{cases} A \setminus X = X \setminus B, \\ X \setminus A = C \setminus X; \end{cases}$

(c)

$\begin{cases} A \cap X = B \setminus X, \\ C \cup X = X \setminus A. \end{cases}$

For which $A$ , $B$ , and $C$ do these systems have a solution?

Problem I.1.33

(a)(b)(c)

Prove that every set is:

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(a)

the union of all its subsets;

(b)

the union of all its finite subsets;

(c)

the union of all its singleton subsets.

Problem I.1.34

Suppose we have a sequence of sets

X_{0} \supseteq X_{1} \supseteq X_{2} \supseteq \ldots \supseteq X_{n} \supseteq \ldots

Prove that the intersection of any infinite subsequence of these sets coincides with the intersection of the whole sequence.

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Problem I.1.35

Suppose we have a sequence of sets

X_{0} \subseteq X_{1} \subseteq X_{2} \subseteq \ldots \subseteq X_{n} \subseteq \ldots

Prove that the union of any infinite subsequence of these sets coincides with the union of the whole sequence.

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Problem I.1.36

Prove the following identities:

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(a)

$\bigcup_{k \in K} \bigcup_{t \in T} A_{kt} = \bigcup_{t \in T} \bigcup_{k \in K} A_{kt}$ ;

(b)

$\bigcap_{k \in K} \bigcap_{t \in T} A_{kt} = \bigcap_{t \in T} \bigcap_{k \in K} A_{kt}$ ;

(c)

$-\left(\bigcup_{k \in K} A_{k}\right) = \bigcap_{k \in K} (-A_{k})$ ;

(d)

$-\left(\bigcap_{k \in K} A_{k}\right) = \bigcup_{k \in K} (-A_{k})$ ;

(e)

$\bigcup_{k \in K} A_{k} \cup \bigcup_{k \in K} B_{k} = \bigcup_{k \in K} (A_{k} \cup B_{k})$ ;

(f)

$\bigcup_{k \in K} (B \cap A_{k}) = B \cap \left(\bigcup_{k \in K} A_{k}\right)$ ;

(g)

$\bigcap_{k \in K} (B \cup A_{k}) = B \cup \left(\bigcap_{k \in K} A_{k}\right)$ .

Problem I.1.37

(a)(b)

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(a)

Prove that for any $K$ , $T$ , $A_{kt}$

\bigcup _{k \in K} \bigcap _{t \in T} A_{kt} \subseteq \bigcap _{t \in T} \bigcup _{k \in K} A_{kt}.

(b)

Prove that in the assertion (a) the inclusion cannot be replaced by equality.

Problem I.1.38

(a)(b)(c)

Prove that:

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(a)

if $A_{t} \subseteq B$ for all $t \in T$ , then $\bigcup_{t \in T} A_{t} \subseteq B$ ;

(b)

if $B \subseteq A_{t}$ for all $t \in T$ , then $B \subseteq \bigcap_{t \in T} A_{t}$ ;

(c)

if $A_{t} \subseteq B_{t}$ for all $t \in T$ , then $\bigcup_{t \in T} A_{t} \subseteq \bigcup_{t \in T} B_{t}$ and $\bigcap_{t \in T} A_{t} \subseteq \bigcap_{t \in T} B_{t}$ .

Problem I.1.39

(a)(b)

Prove that:

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(a)

$\bigcup_{t \in T} A_{t}$ is the smallest set containing all the sets $A_{t}$ ;

(b)

$\bigcap_{t \in T} A_{t}$ is the largest set contained in all the sets $A_{t}$ .

Problem I.1.40

Prove that if $\left(\bigcap_{n \in \mathbb {N} \setminus \left\{ 0\right\} } A_{n}\right) \cap \left(\bigcap_{n \in \mathbb {N} \setminus \left\{ 0\right\} } B_{n}\right) = \emptyset$ , then

\bigcap _{n \in \mathbb {N} \setminus \left\{ 0\right\} } A_{n} \subseteq \bigcup _{n \in \mathbb {N} \setminus \left\{ 0\right\} } [A_{n} \cap (B_{n-1} \setminus B_{n})],

where

\left(\bigcup _{n \in \mathbb {N} \setminus \left\{ 0\right\} } A_{n}\right) \cup \left(\bigcup _{n \in \mathbb {N} \setminus \left\{ 0\right\} } B_{n}\right) \subseteq B_{0}.

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Problem I.1.41

Prove that for any system of sets $A_{0}, \ldots , A_{n}, \ldots$ there exists a system of pairwise disjoint sets $B_{0}, \ldots , B_{n}, \ldots$ such that $\bigcup_{n \in \mathbb {N}} A_{n} = \bigcup_{n \in \mathbb {N}} B_{n}$ and $B_{n} \subseteq A_{n}$ .

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