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Problem: Is the set \( A=\{x(t) \in C[0 ; 1] \mid \forall t \in[0 ; 1], 0

19.7.12 Properties of sets

6.24 $

3) Problem: Show that the field \( \vec{a} \) is potential and find its potential: \[ \vec{a}=\left(\frac{y}{z}+2 x\right) \vec{\imath}+\frac{x}{z} \vec{\jmath}-\frac{x y}{z^{2}} \vec{k} . \]

9.9.2 Potential and solenoidal fields

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Problem: Show that the field \( \vec{a} \) is potential and find its potential: \[ \vec{a}=\left(\frac{y}{z}+z\right) \vec{\imath}+\left(\frac{x}{z}-\frac{1}{y^{2}}\right) \vec{\jmath}+\left(x-\frac{x y}{z^{2}}\right) \vec{k} . \]

9.9.3 Potential and solenoidal fields

1.25 $

Problem: Two finite sets are given: \( A=\{a, b, c\}, B=\{1,2,3,4\} \); binary relations \( P_{1} \subseteq A \times B ; P_{2} \subseteq B^{2} \). Plot \( P_{1}, P_{2} \) graphically. Find \( P=\left(P_{2} \circ P_{1}\right)^{-1} \). Write the domain and the range of all three relation: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check if the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 2),(a, 4),(b, 1),(b, 2),(b, 4),(c, 2),(c, 4)\}, \\ P_{2}=\{(1,1),(2,2),(2,4),(3,3),(3,2),(4,4),(1,3),(4,1)\} . \end{array} \]

6.2.14 Binary relations

3.75 $

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

6.2.15 Binary relations

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Problem: Two finite sets are given: \( A=\{a, b, c\}, B=\{1,2,3,4\} \); binary relations are \( P_{1} \subseteq A \times B ; P_{2} \subseteq B^{2} \). Represent \( P_{1}, P_{2} \) graphically. Find \( P=\left(P_{2} \circ P_{1}\right)^{-1} \). Write the domains of definition and ranges of all three relations: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check if the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 1),(b, 3),(c, 1),(c, 4),(c, 3),(c, 2)\} ; \\ P_{2}=\{(1,1),(1,2),(1,4),(2,1),(2,2),(2,3),(3,3), \\ (3,2),(3,4),(4,3),(4,4),(4,1)\} . \end{array} \]

6.2.16 Binary relations

3.75 $

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

6.2.17 Binary relations

1.75 $

Problem: Two finite sets are given: \( A=\{a, b, c\}, B=\{1,2,3,4\} \); binary relations are \( P_{1} \subseteq A \times B ; P_{2} \subseteq B^{2} \). Represent \( P_{1}, P_{2} \) graphically. Find \( P=\left(P_{2} \circ P_{1}\right)^{-1} \). Write the domains and ranges of all three relations: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to find out if the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 3),(a, 2),(a, 4),(b, 1),(c, 4),(c, 3),(c, 2)\} \\ P_{2}=\{(1,1),(1,4),(2,1),(2,2),(2,4),(3,3),(4,4), \\ (4,3),(3,2),(3,4)\} \end{array} \]

6.2.18 Binary relations

3.75 $

Problem: Calculate the integral using residues: \[ I_{1}=\int_{0}^{2 \pi}\left(\frac{1}{-6-5 \sin x}+\frac{1}{(6-2 \cos x)^{2}}\right) d x \]

10.7.1 Calculating integrals of a real variable using residues

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Problem: Calculate the integral, using residues: \[ I_{2}=\int_{-\infty}^{+\infty} \frac{(4 x-9) \cos \frac{\pi x}{5}+(-8 x-3) \sin \frac{\pi x}{5}}{x^{2}+4 x+40} d x \text {. } \]

10.7.2 Calculating integrals of a real variable using residues

4.99 $

Problem: Calculate the integral, using residues: \[ I=\int_{-\infty}^{+\infty} \frac{3 x^{2}+6 x-4}{\left(x^{2}-8 x+17\right)\left(x^{2}-16 x+68\right)} d x . \]

10.7.3 Calculating integrals of a real variable using residues

2.5 $

Problem: Calculate the integral, using residues: \[ \int_{-\infty}^{+\infty} \frac{d x}{\left(x^{2}+2\right)\left(x^{2}+3\right)^{2}} \text {. } \]

10.7.4 Calculating integrals of a real variable using residues

2.5 $

Problem: Calculate the integral using residues: \[ \int_{0}^{2 \pi} \frac{d x}{4-\sqrt{7} \sin x} \text {. } \]

10.7.5 Calculating integrals of a real variable using residues

2.5 $

Problem: Calculate the integral using residues: \[ \int_{0}^{2 \pi} \frac{d x}{(4+\sqrt{7} \cos x)^{2}} \text {. } \]

10.7.6 Calculating integrals of a real variable using residues

3 $

Problem: Calculate the integral using residues: \[ \int_{-\infty}^{+\infty} \frac{2 x+3}{\left(x^{2}+1\right)^{2}} d x \text {. } \]

10.7.7 Calculating integrals of a real variable using residues

2.5 $

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