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I have some tasks that I want someone could help for me to solve.

Step-by-step solutions would be really good, to really know how these tasks can be done (main purpose is to gain more knowledge in these things):

Letters $C$, $B$ and $D$ mark these facts: $C$ = "R is a student"; $B$ = "G is a sstudent"; $D$ = "P is a student.".

1. Then the fact "There is even one student between these boys" could express by the formula:

A. $\overline{C\& B\& D}$;

B. $C\& B\& D$;

C. $\overline{C\lor B\lor D}$;

D. $C\& B\lor D$;

E. $C\lor B\& D$;

F. $C\lor B\lor D$.

2. The same fact could also be written:

A. $\overline{\overline{C}\& B\& D}$;

B. $\overline{\overline{C}\& \overline{B}\& \overline{D}}$;

C. $\overline{C\& B\& D}$;

D. $C\& B\lor D$;

E. $\overline{C\lor B\lor D}$;

F. $C\lor B\& D$.

3. Formula $\overline{C}\lor \overline{B}\lor \overline{D}$ means the following:

A. Someone, R, G or P (may all) is not a student;

B. Or R, or G is not a student (but not both) and P is not a student;

C. And R, and G is not a student or (but not both) P is not a student;

D. Someone, R, G or P (but not all) is not a student.

4. Function $p(t,s,u)$ is defined the following truth table:

\begin{tabular}{c|c|c|c}t & s & u & p \hline $0$&$0$&$0$&$ 1$ \hline$0$&$0$&$1$&$ 1$ \hline$0$&$1$&$0$&$ 0$ \hline$0$&$1$&$1$&$ 0$ \hline$1$&$0$&$0$&$ 0$ \hline$1$&$0$&$1$&$ 0$ \hline$1$&$1$&$0$&$ 1$ \hline$1$&$1$&$1$&$ 1$ \end{tabular} } Then $p^*(t,s,u)=$:

A. \begin{tabular}{|c|}$p^*$ \hline $ 1$ \hline$ 0$ \hline$ 1$ \hline$ 0$ \hline$ 0$ \hline$ 1$ \hline$ 1$ \hline$ 0$ \end{tabular};

B. \begin{tabular}{|c|}$p^*$ \hline $ 1$ \hline$ 1$ \hline$ 0$ \hline$ 0$ \hline$ 0$ \hline$ 0$ \hline$ 1$ \hline$ 1$ \end{tabular};

C. \begin{tabular}{|c|}$p^*$ \hline $ 1$ \hline$ 0$ \hline$ 1$ \hline$ 0$ \hline$ 1$ \hline$ 1$ \hline$ 0$ \hline$ 0$ \end{tabular};

D. \begin{tabular}{|c|}$p^*$ \hline $ 0$ \hline$ 0$ \hline$ 1$ \hline$ 1$ \hline$ 1$ \hline$ 1$ \hline$ 0$ \hline$ 0$ \end{tabular}.

5. Which fact is correct?

1) $p(t,s,u)=\left(p(t,s,u)\right)^*$;

2) $p(t,s,u)=\left(\left(p(t,s,u)\right)^*\right)^*$.

A. None of them;

B. Both ffacts;

C. 2);

D. 1).

Bool function $G(y,s)$ expressed using the formula $\overline{ (\overline{y}\Rightarrow s)\& (y\Rightarrow \overline{s}) }$:

6. Which fact is correct?

1) function $G(y,s)$ does not change zero;

2) function $G(y,s)$ does not change one.

A. Fact 2;

B. Both facts;

C. Fact 1;

D. None of them.

7. Logical equation $G(y,s)=1$ has number of solutions:

A. 2;

B. 1;

C. No solutions;

D. 3;

E. 4.

8. DNF of the function $G(y,s)$ is:

A. $\overline{y}\& \overline{s} \lor y\& \overline{s} \lor y\& s$;

B. $\overline{y}\& \overline{s} \lor y\& \overline{s}$;

C. $\overline{y}\& \overline{s} \lor y\& s$;

D. $\overline{y}\& \overline{s} \lor \overline{y}\& s$.

9. CNF of the function $G(y,s)$ is:

A. $(y \lor \overline{s})\& (\overline{y} \lor \overline{s})$;

B. $(y \lor \overline{s})\& (\overline{y} \lor s)$;

C. $(\overline{y} \lor s)\& (\overline{y} \lor \overline{s})$;

D. $(y \lor \overline{s})$.Functions $\alpha(x,y,z)$, $\beta(x,y,z)$, $\gamma(x,y,z)$ are defined of their truth tables: \begin{tabular}{c|c|c|c|c|c}$x$&$y$&$z$&$\alpha$&$\beta$&$\gamma$\hline $0$&$0$&$0$&$1$&$0$&$0$ $0$&$0$&$1$&$0$&$1$&$1$ $0$&$1$&$0$&$1$&$1$&$1$ $0$&$1$&$1$&$0$&$0$&$0$ $1$&$0$&$0$&$1$&$1$&$1$ $1$&$0$&$1$&$0$&$0$&$0$ $1$&$1$&$0$&$0$&$0$&$0$ $1$&$1$&$1$&$1$&$1$&$1$ \end{tabular} Indicate correct facts:

10. Which function does not change zero and one?

A. None of them;

B. $\alpha$ and $\gamma$;

C. all functions;

D. $\alpha$;

E. $\beta$ and $\gamma$;

F. $\gamma$;

G. $\alpha$ and $\beta$;

H. $\beta$.

11. Which function is self-dual?

A. $\beta$;

B. $\gamma$;

C. None of them;

D. $\alpha$ and $\gamma$;

E. $\alpha$ and $\beta$;

F. all functions;

G. $\alpha$;

H. $\beta$ and $\gamma$.

12. Which function is monotonic?

A. $\alpha$ and $\gamma$;

B. $\gamma$;

C. None of them;

D. $\beta$;

E. $\alpha$ and $\beta$;

F. all functions;

G. $\alpha$;

H. $\beta$ and $\gamma$.

13 Which function has even one fiction variable?

A. $\alpha$;

B. $\alpha$ and $\gamma$;

C. $\beta$;

D. $\gamma$;

E. None of them;

F. all functions;

G. $\beta$ and $\gamma$;

H. $\alpha$ and $\beta$.

14. Which function is linear?

A. $\gamma$;

B. $\alpha$ and $\beta$;

C. $\beta$;

D. $\alpha$;

E. $\alpha$ and $\gamma$;

F. all functions;

G. $\beta$ and $\gamma$;

H. None of them.

Thanks for your help, I really appreciate it. :)