Please help with bool algebra and logic things

[PowMath]

Hello! I'm a beginner in discrete math and don't actually know how to solve the bool algebra and logc problems. Sorry for errors in formulas - it's my first post here.
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. :)

 

[HallsofIvy]

What language was this originally in? Things like
"There is even one student between these boys" make no sense at all to me! What is the difference between "students" and "boys"? Are you using "students" to mean both boys and girls and "boys" to mean only male students?

Further, things like
C = "R is a student";
B = "G is a sstudent"; [FONT=MathJax_Math][/FONT]
D = "P is a student." make no sense because we have no idea who or what "R", "G", and "P" are!

 

[Evgeny.Makarov]

Welcome to the forum!

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. $C\lor B\lor D$.

I guess that "between" is supposed to mean "among", but I am not sure about "even". If it means "at least", then the answer is F: R is a student OR G is a student OR P is a student.

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. $C\lor B\& D$.

The answer is B by de Morgan's law.

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.

The answer is A.

4. Function $p(t,s,u)$ is defined the following truth table:
\(\displaystyle \begin{array}{c|c|c|c}
t & s & u & p\\
\hline
0&0&0& 1\\
\hline
0&0&1& 1\\
\hline
0&1&0& 0\\
\hline
0&1&1& 0\\
\hline
1&0&0& 0\\
\hline
1&0&1& 0\\
\hline
1&1&0& 1\\
\hline
1&1&1& 1
\end{array}
\)
Then $p^*(t,s,u)=$:

A. (10100110);
B. (11000011);
C. (10101100);
D. (00111100).

If $f^*$ means the dual function to $f$, i.e., $f(x_1,\ldots,x_n)^*=\overline{f(\bar{x}_1,\ldots,\bar{x}_n)}$, then the vector of values of $f^*$ is obtained from the vector of values of $f$ by reversing it and changing each 0 to 1 and vice versa. So the answer is B.

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 facts;
C. 2);
D. 1).

1) is true for this particular function, i.e., it is self-dual. 2) is true for any function, so the answer is B.

According to the https://mathhelpboards.com/rules/ (rules 8 and 11) you can ask one or two questions in each thread, and you have to show some effort, for example, describe your attempts at solving a problem or describe your difficulties.