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Problem: For the function \( f \), given on the segment \( [a, b] \) 1) prove that it's simple; 2) calculate the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it does exist. \[ a=0, \quad b=1, \quad f(t)=\left[\frac{1}{\sqrt{t}}\right] . \]

19.6.1.9 Lebesgue measure and integration

2.54 $

Problem: By definition, calculate the Lebesgue integral \( \int_{[a, b]} f(t) d t \). \[ a=0, \quad b=3, \quad f(t)=|t-1| . \]

19.6.1.10 Lebesgue measure and integration

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Problem: Let \( f \) be an integrable function on the set \( E, g, h \) are measurable on \( E \). What can be said about the integrability of \( g, h \), if: \( f=g \cdot h, g, h>0 \).

19.6.1.11 Lebesgue measure and integration

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Problem: Calculate the Lebesgue integral \( \int_{[0,1]} f(x) d \mu \) from the function \[ f(x)=\left\{\begin{array}{c} x \cdot e^{x}, \quad x \in[0,1] \backslash Q \\ \tan ^{-1} \frac{1}{x+1}, \quad x \in Q \cap[0,1] \end{array}\right. \]

19.6.1.12 Lebesgue measure and integration

3.81 $

Problem: Let \( f \) be an integrable function on \( X \). Is it true that if \( \int_{X} f(t) d \mu=0 \), then \( f(t)=0 \) almost everywhere?

19.6.1.13 Lebesgue measure and integration

3.05 $

Problem: Find the gradient of scalar field \[ u(x, y, z)=y \sin z-x \cos y+x z-2 z^{2}+7 \] and determine whether this field is harmonic.

2.3.24 Gradient and directional derivative

1.02 $

Problem: For given scalar fields \( u, v \) find: a) the derivative of \( u \) in the direction of \( \vec{e} \) at point \( M \), b) the angle between gradients \( u \) and \( v \) at points \( N \) and \( M \), where \[ \begin{array}{l} v=6 \sqrt{6} x^{3}-6 \sqrt{6} y^{3}+2 z^{3}, \quad u=\frac{x z^{2}}{y} \\ M(1 ; 1 ; \sqrt{6}), \quad N\left(\frac{1}{\sqrt{6}} ; \frac{1}{\sqrt{6}} ; 1\right), \quad \vec{e}=(2 ; 3 ; 4) . \end{array} \]

2.3.25 Gradient and directional derivative

1.78 $

Problem: Expand function \( f(x) \) into Fourier series in \( [-\pi ; 2) \) and plot the graph of function \( y=f(x) \) : \[ f(x)=\left\{\begin{array}{cc} 2 x, & -\pi \leq x \leq 0 \\ \frac{\pi}{2}, & 0

2.6.2.21 Trigonometric Fourier series

3.05 $

Problem: The binary relation \( P \); find its domain and range. Check by the definition whether the relation \( P \) is reflexive, symmetric, antisymmetric, transitive. \( P \subseteq \) \( \subseteq R^{2}, P=\{(x, y) \mid x \cdot y>1\} \).

6.2.19 Binary relations

1.27 $

Problem: 1. From the table for the Boolean function \( g\left(x_{1}, x_{2}, x_{3}\right) \) write down the Zhegalkin polynomial, perfect DNF and CNF. 2. Draw the Boolean function \( g\left(x_{1}, x_{2}, x_{3}\right) \) on a Boolean cube and find the minimal DNF. 3. Find the minimal DNF for function \( f\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \). 4. Examine the completeness of the system \( \{f, g\} \). If the system is not complete, add such a function \( h \), that the system \( \{f, g, h\} \) is complete. 5. Implement functions using a circuit of functional elements in the basis \( \left\{\wedge, \mathrm{v}^{-}\right\} \). 6. Using the functions of one of the bases of 4-th option, express functions \( 0,1, z, \bar{x}, x_{1} x_{2}, x_{1} \vee x_{2}, x_{1} \oplus x_{2} \). Implement them on the same basis with a circuit of functional elements. 7. Construct a contact circuit for the function \( g\left(x_{1}, x_{2}, x_{3}\right) \), presenting it in the form of DNF and CNF. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=(0000011001000111), \\ g\left(x_{1}, x_{2} x_{3}\right)=\left(\overline{x_{1}} x_{2} \rightarrow x_{3}\right) \oplus\left(\overline{x_{3}} \leftrightarrow x_{1}\right) . \end{array} \]

6.3.19 Boolean algebra

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Problem: This formula can be converted into PDNF in two ways: 1 ) using the truth table, 2 ) by transformation. a) \( \bar{x} \vee z \), b) \( \overline{(x \vee y)(\bar{x} \vee z)} \).

6.3.20 Boolean algebra

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Problem: From the formula go to the truth table and use the table to construct the perfect CNF. \[ \bar{x} \vee \overline{(y z)} x \text {. } \]

6.3.21 Boolean algebra

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Problem: Convert the formula to Zhegalkin polynomial. \[ (x \overline{y z}) \mid(x \mid(y \mid z)) \text {. } \]

6.3.22 Boolean algebra

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Problem: Identify dummy variables in functions, defined by formulas and express them in a formula containing only essential variables. \[ (x y \vee \bar{x} \bar{y}) \oplus \bar{x} . \]

6.3.23 Boolean algebra

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Problem: A digraph is defined by an adjacency matrix. It is necessary to: a) plot the graph; b) identify components of strong connectivity; c) replace all arcs with edges and find an Euler chain (or cycle) in the obtained undirected graph. \[ \left(\begin{array}{llllll} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 1 \end{array}\right) \]

6.4.28 Graph theory

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