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Problem: It's known that \( 0

12.3.1.3 Algebraic

3.75 $

Problem: Convert the following formula to CNF: \[ f(p, q, t)=(q \rightarrow(p \rightarrow t)) \rightarrow((q \rightarrow \neg t) \rightarrow(q \rightarrow \neg p)) . \]

6.3.4 Boolean algebra

1.25 $

Problem: Prove the logical identity: \[ \overline{A \vee B}=\bar{A} \wedge \bar{B} \text {. } \]

6.3.5 Boolean algebra

0 $

Problem: Let's show that for any sets \( X, Y, Z \) the equalities are true: a) \( X \times(Y \cup Z)=(X \times Y) \cup(X \times Z) \), b) \( (X \cap Y) \times Z=(X \times Y) \cap(Y \times Z) \)

16.5 Topology

1 $

Problem: Show that if \( \rho(x, y) \) is the metric on the space \( X \), then \( \tan ^{-1} \rho(x, y) \) is also a metric on \( X \).

16.6 Topology

2 $

Problem: Let \( X \) is an infinite set, and \( \mathcal{T} \) is the set of those subsets of \( X \), the complement of which is finite. Show that \( \mathcal{T} \) is a topology.

16.7 Topology

4.99 $

Problem: Show some subgroups \( \mathbb{R} \) and \( \mathbb{R}^{n} \).

16.8 Topology

1.25 $

Problem: Let's prove the relation \( X \times Y=A \times B \Leftrightarrow X=A \), \( Y=B \).

16.9 Topology

0 $

Problem: Show that if \( \rho(x, y) \) is a metric on the space \( X \), then \( \frac{\rho(x, y)}{1+\rho(x, y)} \) is also a metric on \( X \).

16.10 Topology

2 $

Problem: Show that there exists a countable subspace \( \mathbb{R}^{2} \), which is a discrete space.

16.11 Topology

2.5 $

Problem: Show that for any \( a \in \mathbb{Z} \) the set \( \mathbb{Z}_{a}=\{z \in \mathbb{Z}: z \) is divisible by \( a\} \) is a subgroup of \( \mathbb{Z} \).

16.12 Topology

1.25 $

Problem: Find the particular solution of the system of linear differential equations: \[ \left\{\begin{array}{l} \dot{x}=\frac{x}{x+3 y} \\ \dot{y}=\frac{3 y}{x+3 y} \end{array} \quad x(1)=y(1)=1 .\right. \]

8.3.7 Systems of ordinary differential equations

1.5 $

Problem: Find the general solution of the linear system: \[ \left\{\begin{array}{c} \dot{x}=4 x+y-e^{2 t} \\ \dot{y}=y-2 x \end{array}\right. \]

8.3.15 Systems of ordinary differential equations

2.5 $

Problem: Find the general solution of the linear system: \[ \left\{\begin{array}{l} \dot{x}=2 x+y \\ \dot{y}=4 x-y \end{array} .\right. \]

8.3.16 Systems of ordinary differential equations

1.25 $

Problem: Find the general solution of the linear system: \[ \left\{\begin{array}{c} \dot{x}=x+2 y \\ \dot{y}=3 x+6 y \end{array} .\right. \]

8.3.17 Systems of ordinary differential equations

1.25 $

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