Math Problems Vault

Problem 12.1

(1)(2)(3)(4)

Let $f: \mathcal{X} \rightarrow \mathcal{Y}$ be a mapping. Prove that:

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(1)

if $\mathcal{A}_{1} \subseteq \mathcal{A}_{2} \subseteq \mathcal{X}$ , then $f\left(\mathcal{A}_{1}\right) \subseteq f\left(\mathcal{A}_{2}\right)$

(2)

$f\left(\mathcal{A}_{1} \cup \mathcal{A}_{2}\right) = f\left(\mathcal{A}_{1}\right) \cup f\left(\mathcal{A}_{2}\right)$

(3)

if $\mathcal{B}_{1}, \mathcal{B}_{2} \subseteq f(\mathcal{X}) \subseteq \mathcal{Y}$ , then $f^{ - 1}\left(\mathcal{B}_{1} \cap \mathcal{B}_{2}\right) = f^{ - 1}\left(\mathcal{B}_{1}\right) \cap f^{ - 1}\left(\mathcal{B}_{2}\right)$

(4)

$f\left(\mathcal{A}_{1} \cap \mathcal{A}_{2}\right) \subseteq f\left(\mathcal{A}_{1}\right) \cap f\left(\mathcal{A}_{2}\right)$ . Can the image of an intersection fail to coincide with the intersection of the images?

Problem 12.2

In how many ways can a one-to-one correspondence be established between two sets each containing $n$ elements?

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Problem 12.3

(1)(2)(3)

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(1)

How many transformations exist of a set consisting of $n$ elements? How many of them are one-to-one?

(2)

How many possibilities are there for the sets of values of transformations of a set of $n$ elements?

(3)

How many mappings exist of a set of $m$ elements into a set of $n$ elements?

Problem 12.4

(1)(2)

Let $f: \mathcal{X} \rightarrow \mathcal{Y}, \left|\mathcal{X}\right| = m, \left|\mathcal{Y}\right| = n$ . Can the mapping $f$ be:

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(1)

A surjection when $n > m$ ?

(2)

An injection when $n < m$ ?

Problem 12.5

(1)(2)(3)

Let $f: \mathcal{X} \rightarrow \mathcal{Y}$ , where $\mathcal{X}, \mathcal{Y}$ are finite sets consisting of the same number of elements. Prove the equivalence of the following statements:

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(1)

$f$ is an injection

(2)

$f$ is a surjection

(3)

$f$ is one-to-one.

Problem 12.6

(1)(2)

Give examples of mappings $f: \mathcal{X} \rightarrow \mathcal{Y}$ of infinite sets $\mathcal{X}, \mathcal{Y}$ such that:

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(1)

$f$ is a surjection, but not an injection

(2)

$f$ is an injection, but not one-to-one.

Problem 12.7

Establish one-to-one correspondences between the set of all natural numbers and the given set:

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(1)

the set of all integers

(2)

the set of all even numbers

(3)

the set of all rational numbers

(4)

the set of all points of the plane whose coordinates are rational (rational points)

(5)

the set of all intervals on the line with rational endpoints

(6)

the set of all disks in the plane with centers at rational points and rational radii

(7)

the set of all polynomials $p(x) = a_{0} + a_{1} x + \ldots + a_{n} x^{n} (n = 0, 1, 2, \ldots )$ with integer coefficients $a_{i} \quad (i = 0, 1, \ldots , n)$ .

Problem 12.8

(1)(2)

Prove that:

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(1)

no one-to-one correspondence can be established between the set of all integers and the set of all sequences of the numbers 0 and 1

(2)

there exists a one-to-one correspondence between the set of all real numbers and the set of all sequences of the numbers 0 and 1.

Problem 12.9

(1)(2)

Let $\mathcal{X}, \mathcal{Y}$ be infinite sets, $f: \mathcal{X} \rightarrow \mathcal{Y}$ a one-to-one mapping, and $\mathcal{Y} \cap \mathcal{Z} = \varnothing$ . Devise a one-to-one mapping of $\mathcal{X}$ onto $\mathcal{Y} \cup \mathcal{Z}$ , if:

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(1)

$\mathcal{Z}$ is finite

(2)

$\mathcal{Z}$ is countable.

Problem 12.10

Prove that any mapping $f: \mathcal{X} \rightarrow \mathcal{Y}$ has at most one inverse mapping.

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Problem 12.11

(1)(2)

Devise a one-to-one mapping of the line:

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(1)

onto the interval $(- 1, 1)$

(2)

onto the segment $\left[ - 1, 1\right]$ .

Problem 12.12

(1)(2)(3)

Devise a transformation of the plane that maps the plane one-to-one:

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(1)

onto the open disk $x^{2} + y^{2} < 1$

(2)

onto the closed disk $x^{2} + y^{2} \leqslant 1$

(3)

onto the square $\left|x\right| < 1, \left|y\right| < 1$ (the coordinate system is rectangular).

Problem 12.13

(1)(2)(3)

Given a linear transformation of the number line: $f(x) = ax + b$ ( $a, b$ are real numbers). Prove that:

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(1)

$f$ is one-to-one if and only if $a \neq 0$

(2)

$f$ preserves the direction of vectors on the line when $a > 0$ and reverses it when $a < 0$

(3)

when $a \neq 0$ , the image of an interval of length $l$ is an interval of length $\left|a\right| l$ .

Problem 12.14

(1)(2)

Given the transformation $f(x) = ax + b$ of the number line. Find:

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(1)

the fixed points of the transformation $f$

(2)

the transformation inverse to the transformation $f(a \neq 0)$ .

Problem 12.15

Write the formula defining the linear mapping of the interval $(a, b)$ onto the interval $(c, d)$ of the number line.

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Problem 12.16

Given linear transformations $f$ and $g$ of the number line: $f(x) = ax + b, g(x) = cx + d$ . Find the products $fg$ and $gf$ . In what case is $fg = gf$ ?

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Problem 12.17

(1)(2)(3)

A mapping $f$ of the number line into the plane is given, in a rectangular coordinate system, by the formulas: $x = a\cos t$ , $y = b\sin t\quad (a > 0, \quad b > 0)$ .

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(1)

Find the image $\mathcal{S}$ of the line under the mapping $f$ .

(2)

Is the mapping $f$ an injection?

(3)

Indicate some sets on the line that are mapped one-to-one onto $\mathcal{S}$ .

Problem 12.18

(1)(2)(3)

A mapping $f$ of the number line into the plane is given, in a rectangular coordinate system, by the formulas: $x = - \operatorname {ch} t$ , $y = \operatorname {sh} t$ .

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(1)

Find the image $\mathcal{S}$ of the line under the mapping $f$ .

(2)

Is the mapping $f$ an injection?

(3)

Find the preimage $t$ of each point of $\mathcal{S}$ .

Problem 12.19

(1)(2)

A transformation $f$ of the plane is given, in a rectangular coordinate system, by the formulas: $x^{ * } = x^{2} - y^{2}, y^{ * } = 2xy$ .

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(1)

Is the transformation $f:$

(1) a surjection,

(2) one-to-one?

(2)

Find the complete preimage of an arbitrary point $\left(x^{ * }, y^{ * }\right)$ of the plane.

Problem 12.20

(1)(2)(3)

A transformation $f$ of the plane is given, in a rectangular coordinate system, by the formulas: $x^{ * } = e^{x} \cos y, y^{ * } = e^{x} \sin y$ .

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(1)

Is the transformation $f$ one-to-one?

(2)

Identify regions in the plane on which $f$ is one-to-one.

(3)

Let $\bar{f}$ be the restriction of the transformation $f$ to the strip $0 < y < \pi$ . Find the formulas defining the mapping inverse to $\bar{f}$ .

Problem 12.21

Given mappings $f: \mathcal{X} \rightarrow \mathcal{Y}, g: \mathcal{Y} \rightarrow \mathcal{Z}$ and $h: \mathcal{Z} \rightarrow \mathcal{U}$ . Prove the associativity of composition of mappings, i.e., the equality $h(gf) = (hg) f$ .

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Problem 12.22

Let $\mathcal{X}, \mathcal{Y}$ be nonempty sets, $\mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ . The mapping $\pi : \mathcal{Z} \rightarrow \mathcal{Y}$ (the projection of $\mathcal{Z}$ onto $\mathcal{Y}$ ) is defined by the equality $\pi ((x, y)) = y$ . Show that $\pi$ is a surjection.

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Problem 12.23

Show that for every set $\mathcal{X}$ there exists an injection $\delta : \mathcal{X} \rightarrow \mathcal{X} \times \mathcal{X}$ .

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Problem 12.24

(1)(2)(3)(4)

The graph of a mapping $f: \mathcal{X} \rightarrow \mathcal{Y}$ is the subset $\Gamma = \left\{ (x, f(x)) \mid x\in \mathcal{X}\right\} \subseteq \mathcal{X} \times \mathcal{Y}$ .

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(1)

Find the image of the set $\mathcal{X}$ under the mapping $\varphi : \mathcal{X} \rightarrow \mathcal{X} \times \mathcal{Y}$ , defined by the equality $\varphi (x) = (x, f(x))$ .

(2)

Prove that $f = \pi \varphi$ (for the definition of $\pi$ , see 12. 22).

(3)

Prove that the mapping $f$ is an injection if and only if $\varphi$ is an injection.

(4)

Prove that $f$ is a surjection if and only if $\pi (\Gamma ) = \mathcal{Y}$ .