 Discrete math
ANSWERED ### If $$\displaystyle{x}_{{{1}}}=-{1},\ {x}_{{{2}}}={1},\ {X}_{{{n}}}={3}{X}_{{{\left({n}-{1}\right)}}}-{\mid}{2}{X}_{{{\left({n}-{2}\right)}}},\ \forall{n}\geq{3}.$$ Find the general term $$\displaystyle{X}_{{{n}}}$$

Discrete math
ANSWERED ### This is a discrete math (combinatorics and discrete probability) problem. Please explain each step in detail and do not copy solutions from Chegg. Consider the random process where a fair coin will be repeatedly flipped until the sequence TTH or THH appears. What is the probability that the sequence THH will be seen first? Explicitly state the value of this probability and walk through the development of the calculation that lead to this value.

Discrete math
ANSWERED ### Find the number of edges in a circulant graph Circ$$\displaystyle{\left[{n},{\left\lbrace{k},{l}\right\rbrace}\right]}$$. In mathematica you can explore these graphs by using the comand CirculantGraph $$\displaystyle{\left[{n},{\left\lbrace{k},{l}\right\rbrace}\right.}$$.

Discrete math
ANSWERED ### Let $$\displaystyle{A}_{{{2}}}$$ be the set of all multiples of 2 except for 2. Let $$\displaystyle{A}_{{{3}}}$$ be the set of all multiples of 3 except for 3. And so on, so that $$\displaystyle{A}_{{{n}}}$$ is the set of all multiples of n except for n, for any $$\displaystyle{n}\geq{2}$$. Describe (in words) the set $$\displaystyle\overline{{{A}_{{{2}}}\cup{A}_{{{3}}}\cup{A}_{{{4}}}\cup\cdots}}$$

Discrete math
ANSWERED ### Function Relation (Discrete math) Let $$\displaystyle{A}={\left\lbrace{0},{1},{2}\right\rbrace}$$ and $$\displaystyle{r}={\left\lbrace{\left({0},{0}\right)},{\left({1},{1}\right)},{\left({2},{2}\right)}\right\rbrace}$$ Show that r is an equivalence relation on A.

Discrete math
ANSWERED ### Discrete math Question. Suppose your friend makes the following English statement "If $$X \oplus Y$$, but $$\displaystyle\sim{X}$$, then we have Y." Convert it into a statement form. Then show that your friend's statement is valid. Is it true that "$$\displaystyle{X}\oplus{Y}$$, but $$\displaystyle\sim{X}$$" is equivalent to Y?

Discrete math
ANSWERED ### Let $$\displaystyle{A}={\left\lbrace{a},\ {b},\ {c},\ {d},\ {e},\ {f}\right\rbrace}.$$ Define the relation $$\displaystyle{R}={\left\lbrace{\left({a},{a}\right)},{\left({a},{c}\right)},{\left({b},{d}\right)},{\left({c},{d}\right)},{\left({c},{a}\right)},{\left({c},{c}\right)},{\left({d},{d}\right)},{\left({e},{f}\right)},{\left({f},{e}\right)}\right\rbrace}$$ on A. a) Find the smallest reflexive relation $$\displaystyle{R}_{{{1}}}$$ such that $$\displaystyle{R}\subset{R}_{{{1}}}$$. b) Find the smallest symmetric relation $$\displaystyle{R}_{{{2}}}$$ such that $$\displaystyle{R}\subset{R}_{{{2}}}$$ c) Find the smallest transitive relation $$\displaystyle{R}_{{{3}}}$$ such that $$\displaystyle{R}\subset{R}_{{{3}}}$$.

Discrete math
ANSWERED ### What is the coefficeint of $$\displaystyle{a}^{{{2}}}{b}^{{{3}}}{c}^{{{4}}}$$ in the expansion of $$\displaystyle{\left({a}+{2}{b}+{3}{c}\right)}^{{{9}}}$$?

Discrete math
ANSWERED ### Prove that any group of 20 people will contain at least one pair of people with the same amount of friends within the group. (Here, you can let $$\displaystyle{S}={\left\lbrace{p}_{{{1}}},{p}_{{{2}}},\ldots,{p}_{{{20}}}\right\rbrace}$$ be an arbitrary set of 20 people, and define $$\displaystyle{n}{\left({p}_{{{i}}}\right)}$$ for $$\displaystyle{1}\leq{i}\leq{20}$$ to be the number of friends for person $$\displaystyle{p}_{{{i}}}$$ within this group. Assume friendship is symmetric, so if someone has 0 friends in the group then there can't be someone with 19 friends. Similarly, if someone has 19 friends in the group then there can't be anyone with 0 friends in the group. What are the possible values of $$\displaystyle{n}{\left({p}_{{{i}}}\right)}.?{)}$$.

Discrete math
ANSWERED ### Discrete Math Question Prove the following statement: "The sum of any two rational numbers is rational."

Discrete math
ANSWERED ### Determine whether the following set equivalence is true $$\displaystyle{\left({A}\cup{B}\right)}\ {\left({A}\cap{C}\right)}={B}\cup{\left({A}\ {C}\right)}$$

Discrete math
ANSWERED ### Discrete Math Prove that Z has no smallest element.

Discrete math
ANSWERED ### For each of the following sets A,B prove or disprove whether $$\displaystyle{A}\subseteq{B}$$ and $$\displaystyle{B}\subseteq{A}$$ a) $$\displaystyle{A}={\left\lbrace{x}\in{Z}:\exists_{{{y}\in{z}}}{x}={4}{y}+{1}\right\rbrace}$$ $$\displaystyle{B}={\left\lbrace{x}\in{Z}:\exists_{{{y}\in{z}}}{x}={8}{y}-{7}\right\rbrace}$$

Discrete math
ANSWERED ### Solve the recursion: $$\displaystyle{A}_{{{1}}}={1}.\ {A}_{{{2}}}=-{1}$$ $$\displaystyle{A}_{{{k}}}={5}{A}_{{{k}-{1}}}-{6}{A}_{{{k}-{2}}}$$

Discrete math
ANSWERED ### This is for Discrete Math. Three horses A,B and C, can finish a race in how many ways? (ties are possible) 1) 10 2) 15 3) 9 4) 12 5) 13

Discrete math
ANSWERED ### Discrete Mathematics a) Write the proposition for the following English sentence: 1) a) If Benjamin have cough, fever, and no smell, Benjamin is COVID19 positive. b) Write inverse, converse and countrapositive of English sentence in (A) 2) Let p and q be the propositions "The election is decided" and "The votes have been counted," respectively. Express each of these compound propositions as English sentences. b) Proof using laws of logic 1) $$\displaystyle{\left({p}\wedge{q}\right)}\rightarrow{p}$$ is tautology 2) $$\urcorner p\leftrightarrow q\Leftrightarrow p\leftrightarrow\urcorner q$$

Discrete math
ANSWERED ### Finding a Cartesian Product. Let $$\displaystyle{A}_{{{1}}}={\left\lbrace{x},{y}\right\rbrace},\ {A}_{{{2}}}={\left\lbrace{1},{2},{3}\right\rbrace},$$ and $$\displaystyle{A}_{{{3}}}={\left\lbrace{a},{b}\right\rbrace}.$$ a) Find $$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}.$$ b) Find $$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}\times{A}_{{{3}}}.$$

Discrete math
ANSWERED ### Prove ar disprove: $$\displaystyle{\left\langle{\mathbb{{{Z}}}}_{{{4}}},\oplus_{{{4}}},{0}\right\rangle}\stackrel{\sim}{=}{\left\langle{B}_{{{2}}},+,{00}\right\rangle}$$ where $$\displaystyle{\left(+\right)}$$ is the Boolean (bitwise) sum on $$\displaystyle{B}_{{{2}}}$$

Discrete math
ANSWERED ### Let R be a relation on $$\displaystyle{\mathbb{{{Z}}}}$$ defined by $$\displaystyle{R}={\left\lbrace{\left({p},{q}\right)}\in{\mathbb{{{Z}}}}\times{\mathbb{{{Z}}}}{\mid}{p}-{q}\right.}$$ is a multiple of $$\displaystyle{3}\rbrace$$ a) Show that R is reflexive. b) Show that R is symmetric. c) Show that R is transitive.
ANSWERED 