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Problem: a) Bring an example of an Eulerian graph with an even number of vertices which does not have a perfect matching. b) Prove that any Hamiltonian graph with an even number of vertices has a perfect matching.

6.4.3 Graph theory

5.18 $

Problem: Reduce the predicate logic formula to prenex normal form. Is the formula on the set \( M=\{1,2\} \) : 1) satisfiable, 2) refutable, 3) generally valid, 4) unsatisfiable? Calculate the truth value of the formula on the set \( M \) with the following predicates: \begin{tabular}{|c|c|c|} \hline\( x \) & 1 & 2 \\ \hline\( P(x) \) & 1 & 0 \\ \hline\( R(x) \) & 0 & 1 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline\( Q(x, y) \) & 1 & 2 \\ \hline 1 & 1 & 0 \\ \hline 2 & 0 & 0 \\ \hline \end{tabular} \[ \forall x P(x) \rightarrow(R(x) \rightarrow \exists y Q(x, y)) \text {. } \]

6.5.1 Predicate calculus

3.11 $

Problem: Establish the truth of the logical expression using two methods: 1) by defining quantifiers and 2) by the method of concretization. a) \( \forall x \exists y(A(y) \vee B(x))=\exists x A(x) \vee \forall x B(x) \), b) \( \forall x \exists y P(x, y) \Rightarrow \exists x \exists y P(x, y) \).

6.5.2 Predicate calculus

4.66 $

Problem: Determine the truth values of formulas in a given interpretation: a) \( \exists x \forall y(P(x) \Rightarrow Q(y)) \), b) \( \exists x(P(x) \& Q(x)) \) \[ D=\{a, b\} \] \begin{tabular}{|c|c|c|} \hline\( x \) & \( P(x) \) & \( Q(x) \) \\ \hline\( a \) & 1 & 0 \\ \hline\( b \) & 0 & 1 \\ \hline \end{tabular}

6.5.3 Predicate calculus

1.29 $

Problem: The city council consists of a mayor and six elders. How many different four-member commissions can be formed if the mayor is in each commission? What if the mayor is not a member of any commission?

6.6.1 Combinatorics

1.55 $

Problem: In how many ways can six people choose from six gloves, right and left, so that none of them gets a pair?

6.6.2 Combinatorics

2.07 $

Problem: How many different ten-digit numbers can be written using these three digits \( 1,2,3 \), if digit 3 is used exactly twice in each number? How many of these numbers are divisible by 9 ?

6.6.3 Combinatorics

3.11 $

Problem: Determine the time and capacity complexity of algorithms for solving the following problems under uniform and logarithmic weight criteria. Calculate the average of the elements of an array of \( n \) numbers.

6.7.1 Theory of algorithms and formal languages

3.11 $

Problem: Check whether the word abababa is a theorem in palindrome calculus.

6.7.2 Theory of algorithms and formal languages

1.81 $

Problem: Prove the identity using the mutual inclusion method and Euler-Venn diagrams. \[ \begin{array}{l} A \cup(B \cap C)=(A \cup B) \cap(A \cup C), \\ (A \cup B) \cdot C=(A \cdot C) \cup(B \cdot C) . \end{array} \]

6.8.1 Set theory

3.88 $

Problem: Write down the set in gray.

6.8.2 Set theory

0.52 $

Problem: Prove that: \[ \left(A_{1} \cap \ldots \cap A_{n}\right)-\left(B_{1} \cap \ldots \cap B_{n}\right) \subseteq\left(A_{1} \doteq B_{1}\right) \cup \ldots \cup\left(A_{n} \doteq B_{n}\right) \text {. } \]

6.8.3 Set theory

1.81 $

Problem: Prove that: \[ U^{2} \backslash(A \times B)=[(U \backslash A) \times U] \cup[U \times(U \backslash B)] \text {. } \]

6.8.4 Set theory

2.59 $

Problem: Prove that there exists such a maximal (by inclusion) set \( A \subseteq \mathbb{R} \) that \[ \forall a, a^{\prime} \in A, a-a^{\prime} \notin Q \text { when } a \neq a^{\prime} \text {. } \]

6.8.5 Set theory

5.18 $

Problem: Write the canonical equation of the straight line that is the intersection of planes: \( \Gamma x+\mathrm{H} y+3 z=\Gamma \) and \( x-y-\mathrm{Hz}=1 \), where \( \Gamma=\mathrm{H}=2 \).

3.4.4 Tangents and normals

1.81 $

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