 Discrete math
ANSWERED ### Discrete Mathematics Basics 1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where $$\displaystyle{\left({a},{b}\right)}\in{R}$$ if and only if I) everyone who has visited Web page a has also visited Web page b. II) there are no common links found on both Web page a and Web page b. III) there is at least one common link on Web page a and Web page b.

Discrete math
ANSWERED ### Solve the following. a) Find a closed formula for the sequence $$\displaystyle{6},-{12},{18},-{24},{30},-{36},\ldots$$ b) Find the sum $$\displaystyle{\sum_{{{k}={1}}}^{{{4}}}}{\frac{{{1}}}{{{k}!}}}$$.

Discrete math
ANSWERED ### Discrete Math Letting $$\displaystyle{s}_{{{1}}}={0}$$, find a recursive formula for the sequence 0,1,3,7,15,...

Discrete math
ANSWERED ### Prove that all elements in S are multiples of 3 by structural induction. $$\displaystyle{3}\in{S}$$ $$\displaystyle{x},{y}\in{S}\rightarrow{\left({x}+{y}\right)}\in{S}$$

Discrete math
ANSWERED ### An Example with Intervals Let the universal set be the set R of all real numbers and let $$\displaystyle{A}={\left\lbrace{x}\in{R}{\mid}-{1}{<}{x}\leq{0}\right\rbrace}$$ and $$\displaystyle{B}={\left\lbrace{x}\in{R}{\mid}{0}\leq{x}{<}{1}\right\rbrace}.$$ a) Find $$\displaystyle{A}\cup{B}$$ b) Find $$\displaystyle{A}\cap{B}$$ c) Find $$\displaystyle{A}^{{{c}}}$$

Discrete math
ANSWERED ### Prove the following statement using mathematical induction or disapprove by counterexample. If you use mathematical induction, then you should explain each step and you should highlight $$\displaystyle{P}{\left({n}\right)},\ {P}{\left({k}\right)},\ {P}{\left({k}+{1}\right)},$$ the inductive hypothesis, etc. Explaining each step is very important. $$\displaystyle{1}+{5}+{9}+{13}+\cdots+{\left({4}{n}-{3}\right)}=\frac{{{n}{\left({4}{n}-{2}\right)}}}{{2}}$$

Discrete math
ANSWERED ### What if the centipede wants 50 (possibly different) matching pairs, one pair for each pair of feet? Consider first a centipede who only owns green and brown socks, then a centipede who also ownsstripey socks, and then a centipede who owns k different colors of socks.

Discrete math
ANSWERED ### How many elements are in the set { 2,2,2,2 } ?

Discrete math
ANSWERED ### Discrete math Given $$\displaystyle{U}={\left\lbrace{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}\right\rbrace},{S}_{{{1}}}={\left\lbrace{1},{3},{5},{7},{9}\right\rbrace},{S}_{{{2}}}={\left\lbrace{1},{2},{4},{6},{8},{10}\right\rbrace}$$. What is the $$\displaystyle{S}_{{{1}}}\cap{S}_{{{2}}}$$ in bit strings? Select one : 1) 00 2) 01 3) 0000000000 4) 0000000001 5) 1000000000

Discrete math
ANSWERED ### Let G be the graph with vertices $$\displaystyle{v}_{{{1}}},{v}_{{{2}}}$$ and $$\displaystyle{v}_{{{3}}}$$ and the matrix $$\begin{bmatrix}1 & 1 & 2\\ 1 & 0 & 1 \\ 2 & 2 & 0 \end{bmatrix}$$ To find the number of walks of from $$\displaystyle{v}_{{{1}}}$$ to $$\displaystyle{v}_{{{3}}}$$ we need to find matrix $$\displaystyle{A}^{{{2}}}$$

Discrete math
ANSWERED ### 1) $$\displaystyle{110001}$$ binary number is equivalent to which of the following decimal number a) $$\displaystyle{48}$$ b) $$\displaystyle{49}$$ c) $$\displaystyle{59}$$ d) $$\displaystyle{58}$$ 2) Which among the following is the recursive definition of Factorial, i.e., n! ? a) $$\displaystyle{f{{\left({0}\right)}}}={0},\ {f{{\left({n}\right)}}}={\left({n}\right)}{f{{\left({n}-{1}\right)}}},$$ where $$\displaystyle{n}\in{Z}$$ and $$\displaystyle{n}\geq{1}$$ b) $$\displaystyle{f{{\left({0}\right)}}}={1},\ {f{{\left({n}\right)}}}={\left({n}+{1}\right)}{f{{\left({n}-{1}\right)}}},$$ where $$\displaystyle{n}\in{Z}$$ and $$\displaystyle{n}\geq{1}$$ c) $$\displaystyle{f{{\left({0}\right)}}}={1},\ {f{{\left({n}\right)}}}={\left({n}\right)}{f{{\left({n}-{1}\right)}}},$$ where $$\displaystyle{n}\in{Z}$$ and $$\displaystyle{n}\geq{1}$$ b) $$\displaystyle{f{{\left({0}\right)}}}={0},\ {f{{\left({n}\right)}}}={\left({n}+{1}\right)}{f{{\left({n}-{1}\right)}}},$$ where $$\displaystyle{n}\in{Z}$$ and $$\displaystyle{n}\geq{1}$$

Discrete math
ANSWERED ### A question from Discrete Mathematics. Topic Summations. Compute the following summation: $$\displaystyle{\sum_{{{k}={1}}}^{{{2}{k}{\left({1}+{k}\right)}}}}{k}{\left({k}+{1}\right)}$$ $$\displaystyle{k}={1}$$

Upper Level Math
ANSWERED ### As of October, 2016 the world`s largest pumpkin weighed 1190 kilograms. In order to grow a record-setting pumpkin, the fruits mass must increase by an average of $$1.27ft^3/day$$. If we assume that the dinsity of raw pumpkin is approximately $$0.50g/cm^3$$ we can convert this to an increase in volume of $$1.27ft^3/day$$. Question: If the volume of a spherical pumpkinn is increasing at $$\frac{1}{27}ft^3/day$$, then how fast is the surface area of the pumpkin changing when the diameter of the pumpkin is b ft? Give your answer with two decimal places, and ensure that you include unots. Note: Equations for the volume and surface area of a sphere are on D2L in several places.

Discrete math
ANSWERED ### We have a recursively defined sequence $$\displaystyle{a}_{{{n}}}$$. $$\displaystyle{a}_{{{0}}}={0},{a}_{{{1}}}={3}$$, and $$\displaystyle{a}_{{{n}}}={3}{a}_{{{n}-{1}}}-{2}{a}_{{{n}-{2}}}$$ for $$\displaystyle{n}\geq{2}$$ We would like to prove that f or all $$\displaystyle{n}\geq{0},{a}_{{{n}}}={3}\cdot{2}^{{{n}}}-{3}$$. Prove this using the stronger mathematical induction.

Discrete math
ANSWERED ### [Graph Theory - Discrete Mathematics] How do you solve this? Let $$\displaystyle{G}={\left({V},{E}\right)}$$ be a connected simple graph, and call the edge $$\displaystyle{\left({u},{v}\right)}\in{E}$$ essential if the graph obtained by removing (u, v) from G is disconnected. Show that G is a tree if and only if all its edges are essential.

Discrete math
ANSWERED ### What is the cardinality of the set $$\{f\mid f : \rightarrow,\ f$$ is a bijectioon such that $$\displaystyle{f{{\left({i}\right)}}}\ne{}{i},$$ for every $$\displaystyle{i}=\lbrace{1},\ {2},\ {3},\ {4},\ {5},\ {6},\ {7}\rbrace$$?

Discrete math
ANSWERED ### 1) What is the remainder $$\displaystyle{\left({r}\right)}$$ when $$\displaystyle-{43}$$ divided by $$\displaystyle{12}?$$ a) None of the Option b) $$\displaystyle-{5}$$ c) $$\displaystyle{5}$$ d) $$\displaystyle{7}$$ 2) Consider the recursively defunction: $$\displaystyle{f{{\left({0}\right)}}}={2},\ {f{{\left({n}\right)}}}={2}{f{{\left({n}-{1}\right)}}}+{5}$$ where n is an integer greater than 0. The function value at $$\displaystyle{n}={2},\ {i}.{e}.,\ {f{{\left({2}\right)}}}=?$$ a) $$\displaystyle-{23}$$ b) $$\displaystyle{23}$$ c) $$\displaystyle{7}$$ d) $$\displaystyle-{7}$$

Discrete math
ANSWERED ### Discrete mathematics If $$\displaystyle{x}_{{{1}}}={2},{x}_{{{n}}}={4}{X}_{{{n}-{1}}}-{4}{n}\forall{n}\geq{2}$$. Find the general term xn.

Upper Level Math
ANSWERED ### Although tea is the world’s most widely consumed beverage after water, little is known about its nutritional value. Folacin is the only B vitamin present in any significant amount in tea, and recent advances in assay methods have made accurate determination of folacin content feasible. Consider the accompanying data on folacin content for randomly selected specimens of the four leading brands of green tea. $$\begin{array} \\ \text{Brand}& \text{Observations}\\ \hline 1&7.9&6.2&6.6&8.6&8.9&10.1&9.6\\ 2&5.7&7.5&9.8&6.1&8.4\\ 3&6.8&7.5&5.0&7.4&5.3&6.1\\ 4&6.4&7.1&7.9&4.5&5.0&4.0 \\ \hline \end{array}$$ (Data is based on “Folacin Content of Tea,” J. Amer. Dietetic Assoc., 1983: 627–632.) Does this data suggest that true average folacin content is the same for all brands? a. Carry out a test using α=.05α=.05 via the P-value method. b. Assess the plausibility of any assumptions required for your analysis in part (a). c. Perform a multiple comparisons analysis to identify significant differences among brands.
ANSWERED 