Explore Discrete Math Examples

Let A, B, and C be sets. Show that ${\displaystyle (A-B)-C=(A-C)-(B-C)}$

Solve the traveling salsman roblem for this graph by finding the total weight of all circuits and determining a circuit with minimum total weight

a)Let ${\displaystyle A=\{1,2,3,4,5,6\},A_1=\{1,2\},A_2=\{3,4\}\text{ and }{A}_3=\{5,6\}.Is\{A_1,A_2,A_3\}}$ a partition of A?
b) Let Z be the set of all integers and let:
${\displaystyle T_0=\{n\in Z\mid n=3k,\text{ for some integer }k\}}$
${\displaystyle T_1=\{n\in Z\mid n=3k+1,\text{ for some integer }k\}}$ ,and
${\displaystyle T_2=\{n\in Z\mid n=3k+2,\text{ for some integer }k\}}$
Is ${\displaystyle \{T_0,T_1,T_2\}}$ a partition of Z?

Given proposontial variables. Tranlate the following sentences into propositional logic

Select the correct general solution for the recurrence relation given below:
${\displaystyle f_n=10\cdot f_{n-1}-25\cdot f_{n-2}}$
${\displaystyle f_n={\alpha }_1{(-5)}^n+{\alpha }_2{(-5)}^n}$
${\displaystyle f_n={\alpha }_1{(-5)}^n+{\alpha }_{2}n{(-5)}^n}$
${\displaystyle f_n={\alpha }_1{\left(5\right)}^n+{\alpha }_{2}n{(-5)}^n}$
${\displaystyle f_n={\alpha }_1{\left(5\right)}^n+{\alpha }_2{\left(5\right)}^n}$
${\displaystyle f_n={\alpha }_1{\left(5\right)}^n+{\alpha }_2{(-5)}^n}$
${\displaystyle f_n={\alpha }_1{\left(5\right)}^n+{\alpha }_{2}n{\left(5\right)}^n}$

b)Prove that the following code returns the $n^{th}$ odd natural number by structural induction.For instance , func(3)=5.Consider func(0) to be undefined.

Prove the following using propositional logic:

What is the cardinality of the following set?

A)If ${\displaystyle X=\{1,2,3,4,\dots 10\},f_A=0101010101\text{ and }B=1,2,4,67,9.\text{ Find }|A\cdot B|}$
B) Let $A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\end{array}\right]\text{ and }C=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 1\end{array}\right]$
Find ${\displaystyle (B\cdot C)\cdot A}$

Find the discrete Fourier approximation $g_2(x)$ for $f(x)$ based on the table information.
$x|-\frac{\pi }{2}|0|\frac{\pi }{2}|\pi$
$f(x)|0|1|3|-2$

Find the discrete Fourier approximation $g_2(x)$ for $f(x)f(x)$ based on the table information.
$x|0|\pi$
$f(x)|0|2$

If s is a string, define its reverse $s^R$ as follows.
$\begin{array}{|cc|}\hline B.& {\lambda }^R=\lambda \\ R.& \text{If s has one or more symbols, write}\\ & s=ra\\ & \text{where a is a symbol and r is a string (possibly empty). Then}\\ & s^R=(ra{)}^R=a{r}^R.\\ \hline\end{array}$
Compute $(cubs{)}^R$. Justify each step using the definition.

Consider a Poisson process on $[0,\mathrm{\infty })$ with parameter ${\displaystyle \lambda }$ and let T be a random variable independent of the process. Assume T has an exponential distribution with parameter v. Let $N_T$ denote the number of particles in the interval $[0,T]$. Compute the discrete density of $N_T$.