Solved: 8 A bag of chocolates has 20 milk chocolates, 20 dark chocolates and 20 white chocolates. a) What is the minimum number of chocolates you must select at random from the bag to guarantee that you will get at least 5 of the same kind of chocolates? b) What is the minimum number of chocolates you must select at random from the bag to guarantee that you will get at least 5 milk chocolates? 9 Usually, a person doesn’t drink more than 3000 mL of water per day. On a regular day, there are 10,500 people on some university campus. At least how many people on campus would drink the exact same amount of water (in mL) on that day?

How many ways are there to select 9 countries in the United Nations to serve on a council if 1 is selected from a block of 50, 2 are selected from a block of 67 and 6 are selected from the remaining 72 countries?

Out of 300 students taking discrete mathematics, 60 take coffee, 27 take cocoa 36 take tea, 17 take tea only, 47 take chocolate only, 7 take chocolate and cocoa ,3 take chocolate, tea and cocoa ,20 take cocoa only ,2 take tea, coffee …

1) Let P, Q, and R be the propositions P : Grizzly bears have been seen in the area. Q : Hiking is safe on the trail. R : Berries are ripe along the trail. Write these propositions using p, q, and r and logical connectives (includ…

Prove or disprove: every transitive relation on a set X with more 2 points is reflexive

A. Give a brief definition of the following: 1. Graph 2. Degree of a vertex 3. Isomorphic graphs B. Give 4 types of graphs and give a brief description (you may describe in words or just draw a sample graph).

Eulerian and Hamiltonian Graphs, Weighted Graphs A. Define the following: 1. Walk 2. Path, Trail 3. Cycle, Circuit B. What is Eulerian Graph? C. What is Hamiltonian Graph? D. Describe how to solve the Konigsberg Problem.

A. Give a summary of the Greedy Algorithm. B. Give a summary of the Edge-Picking Algorithm. C. What is a graph coloring, and how is it applied?

Construct the Hasse diagram for ({1, 2, 3}, <=)

Let R be the relation on the set A = {a, b, c, d, e, f} and R = {(a,c), (b,d), (c,a), (c,e), (d,b), (d,f), (e,c), (f,d)} Find the transitive closure pf R using Warshall’s algorithm.

Given set S = {1, 2, 3, 4, 5, 6} and a partition of S, A1 = {1, 2, 3} A2 = {4, 5} A3 = {6} Find the ordered pairs that make up the equivalence relation R produced by that partition

Consider following relation on set (1, 2, 3, 4, 5, 6} R= {(i,j) | |i-j|=2} Is R transitive? Is R reflexive? Is R symmetric?

How many bit strings of length 8 either start with 1 bit or end with 2 bits 0

Out of 5 male and 6 female a committee of 5 is to be formed. Find the number of ways in which it can be formed so that among the person chosen in the committee there are: i) exactly 3 males and 2 females ii) at least 2 males and one f…

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements.

Define the relation of d on A by xdy if x is contained within y. For example, 01d101. Draw a digraph for this relation.

In a group of 25 people, there is a ____ chance that at least two of them has shaken hands with the same number of people.

For every integer n≥1,(n3+11n)⋅(8n−14n+27) is divisible by 6 7 42 84

If A={1,2,3,6,12,18,24,30,36,150} and the partial ordering relation is divides then draw the hasse diagram

Let A= {1,2,3,4,5,} and R be a relation on A, such that R = [(1,4), (2,2), (3,4), (3,5), (4,1), (5,2), (5,5)]. Use warshall’s Algorithm to find the matrix of transitive closure of R

Let A= {1,2,3,4,5,6} and R be a relation on A, such that R = [(x, y): |x-y| =2]. Use warshall’s Algorithm to find the matrix of transitive closure of R. and draw its digraph

Let A= {1,2,3,4,6,9,12} let a R b if a is divided by b. Show that R is POSET, Draw Hasse Diagram. Prove or disprove if it is a Lattice.

Draw the graph and its equivalent Hasse diagram for divisibility on the set {1, 2, 3, 6, 12, 24, 36, 48}

a) Let A = (1,2,3,4) and R = ((1 ,2),(2,4),( 1,3 ),(3 ,2)}. Find the transitive closure of R by Warshall’s algorithm. b) Let A = {a,b,c}. show that (P(A),c ) is a poset and draw its Hasse diagram. c) Define Chains and Antichains.

Let R be a binary relation on the set of all positive integers such that R = { (a,b) | a- b is an odd positive integer } Is R reflexive, symmetric, antisymmetric, transitive? Is R an equivalence relation? A partial ordering relation

Function f, g, h are defined on a set X = (1, 2, 3) as f = ((1, 2), (2, 3), (3, 1)} g = {(1,2),(2,1),(3,1)} h = ((1,1), (2,2),(3,1)} i) Find fog, gof. Are they equal? ii) Find fogoh and fohog. Define partial function

In a class has 124 students, of these students 47 are involved in math club, 25 involved in science club and 36 involved in English club so how many are involved in both physics and English club?

Calculate the number of bit strings of length 5 or less.

(2) How many different license plates can be made consisting of 10 characters if the first 5 characters consist of uppercase English letters, while the final 5 characters are digits?

Compute the number of functions f from the set {0,1,2,...; n} (where n is a positive integer), to the set {0,1}.

Calculate the number of one-to-one functions there are from a set with 6 elements to sets with the following numbers of elements: (a) 5 (b) 6 (c) 7 (d) 8

Use the Pigeonhole Principle to show that if there are 31 students in a class, then at least two have first names that begin with the same letter.

Let d be a positive integer. Show that among any group of d+ 1 (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by d.

Assume that a class consists only of students residing the 50 US states. Find the smallest number of students that must be enrolled in a class to guarantee that there are at least 5 students from the same state

8) How many different 6-permutations are there of elements in the set {a, b, c, d, e, f, g, h}?

Compute the number of bit strings of length 14 that contain... (a) exactly four 1s (b) at most four 1s (c) at least four 1s (d) an equal number of 0s and 1s.

Let S be a finite set, with |S|= 200. Find the number of subsets of S containing more than 2 elements

1) Suppose a group contains 7 men and 7 women. How many ways are there to arrange these people in a row if the men and women alternate?

12) Suppose that a room contains 10 cats and 15 dogs. How many ways are there to form a committee consisting of 6 of these animals if it must have more dogs than cats?

Find the number of terms in the complete expansion of (x+y)^150 after like terms are collected together.

Find the coefficient of x^7 y^9 in the expansion of (4x+ 5y)^16.

Let S be a finite nonempty set. Show that the number of subsets of S of even cardinality equals the number of subsets of S of odd cardinality.

Use Mathematical Induction to prove the Binomial Theorem.

Let S={a, b, c, d, e}. Find the number of ways to select 7 elements from S when repetition is allowed. The order in which the elements are chosen does not matter

How many solutions are there to the equation x1+x2+x3+x4+x5+x6= 25 where the xi(for i= 1,2,3,4,5,6) are non-negative integers?

Compute the number of distinct arrangements of the characters ABBCCCDDDDEEEEE

How many bit strings can be formed by using seven 1s and nine 0s?

Which is the following statements is True? Every path is a trail but every walk is not a trail Every walk is a trail but every path is not a trail. Both paths and walks are trails. Neither a path nor a walk is a trail 1 point L…

Q.Which of the following can be a degree sequence of a simple graph? {4,4,3,2,1} {4,3,2,1} {4,4,3,2} {4,2,2,2,2} Q.If G is a connected simple graph and e,f and g are edges of G, then which of the following is/are true: There …

Width of a tree is defined as the largest distance(number of edges in the shortest path) between any two vertices. What is the smallest possible width of the tree with 100 vertices? 1 2 99 101 1 point How many complete graphs w…

Find a counter example to the following universally quantified statements, where the domain for all variables consist of all integers. ∀x, x^2, <=x^3.

given: f(n) = 3f(n/2) + 5n^3 where n=2k and f(1) = 2 find f(n) and a big -0 estimate of f(n)

Given: An=3an-1 - 4an-3 + f(n) find the particular soulution an^(p) when a) f(1) = n2(2)^n b) f(n) = n(3)^n

a) How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen? b) How many must be selected to guarantee that at least three hearts are selected?

Statement: “If n is any even integer, then ” Using Direct proof method

make a truth table (~p∧q) ∨ (p∧~q)

The sum of two even numbers is even. Using Direct proof method

“The product of two odd numbers is odd.” Using Direct proof method

Consider the following sequence of successive numbers of the 2k -th power: 1, 2^2k, 3^2k, 4 ^2k, 5 ^2k, ... Show that the difference between the numbers in this sequence is odd for all k ∈ N.

In a class of 30 students, 11 students like mathematics, 12 students like geography, and 12 students do not like mathematics nor geography. How many students like mathematics and geography

11 integers are arbitrarily chosen. Prove that 2 of them have the same unit digit

There are some people (more than 1 person) in a party. Prove that 2 of them have the same number of friends in the party ( Hint: friendship is a mutual relation)

You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N): λ(n) = 0 , λ(x + 1) = λ(x) , λ (n +1/2)=1 Find two functions p, q : R → R with q(x) not equal to 0 for all x such that λ(x) = q(x)(p(x) + 1).

You have given a function λ:R-> R with the following properties (x ∈ R, n∈ N) : Λ(n) =0, λ(x+1)= λ(x), λ(n+1/2)=1 Find two functions p,q:R-> Rwith q(x) not equal to 0 for all x such that λ(x)= q(x)(p(x)+1)

Let X = {0, 1, 2}. List the elements of each of the following sets: a. {z : z = 2x and x ∈ X} b. {z : z = x + y where x ∈ X and y ∈ X}

Let A = {n ∈ Z | n = 5r for some integer r} and B = {m ∈ Z | m = 20s for some integer s}. a. Is A ⊆ B? Explain. b. Is B ⊆ A? Explain

Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find a. A × B × C. b. C × (B × A).

Which of the following are partitions of , the set of real numbers? Explain your answers. a. {In : n ∈ ℤ}, where In = {x ∈ ℝ : n ≤ x ≤ n + 1} b. {Jn : n ∈ ℤ }, where Jn = {x ∈ ℝ: n ≤ x < n + 1}

For each of the following sets, determine its power set. a. A = {a, b, c, d} b. A = {{1}, {1, 2}}

A professor in a discrete mathematics class passes out a form asking students to check all the mathematics and computer science courses they have recently taken. The finding is that out of a total of 50 students in the class, 30 took …

For the proposition(p ∨¬r)∧(¬p∨(q∨¬r) a. Draw the truth table b. Build the logic circuit which outputs the given compound proposition from input bits p, q, and r.

PDNF of p->((p->q)^~(~qv~p))

Obtain DNF for ~(p → (q ^ r)) and also obtain PCNF for (p → q) ^ (qr) without using truth tables.

in a class of 35 students, each student studies Chemistry or woodwork. Twice as many study chemistry as study woodwork and half as many study only chemistry as study both. illustrate the information on a venn diagram and hence find th…

write the relation on {a,b,c,d} in ordered pairs.

Write each statement in symbolic form, using the given symbols. Let p: I will study. Let q: I will pass the test. Let r: I am foolish. 1. I will study or I am foolish. 2. I will study or I will not pass the test. 3. I will study an…

There is an island with two mythical creatures, the Komos and the Dragos. Komos loves eating visitors to the island while Dragos hates to eat visitors. One fine day, you mysteriously end up on the island and encounter two creatures, B…

Write each statement in symbolic form, using the given symbols. Let p: I will study Let q: I will pass the test Let r: I am foolish 1. I will study or i am foolish 2. I will study or i will not pass the test. 3. I will study and I w…

Let us denote Sn = an + bn + cn for arbitrary numbers a, b, c. It is known that S1 = 8, S2 = 66, S3 = 536 for some values of a, b, c. What is the smallest possible value of S242 — S41 S43?

Let A=[top(1 1) bottom (0 1)] be a matrix consisting of real numbers (we are not interpreting this as a bit matrix). Find a formula for An, where n ϵ Z+, and use Mathematical Induction to prove that your formula is correct.

Let A be a square matrix, and AT denotes the transpose of A. Show that the following hold true. (a) (AT)T=A (b) (A+B)T=AT+BT (c) (AB)T=BT AT.

Show that the relation R=∅ on a nonempty set S is symmetric and transitive, but not reflexive.

Show that the relation R=∅ on the empty set S=∅ is reflexive, symmetric, and transitive.

(a) Find the number of relations on the set S={a, b, c, d, e}? (b) How many relations are there on the set S={a, b, c, d, e} that contain (a, a) and (b, b)?

Let S be a set with 6 elements and let a and b be distinct elements of S. How many relations R are there on S such that... (a) (a, b) ϵ R? b) (a; b) ∉ S? (c) no ordered pair in R has a as its first element? (d) at least one ordered pa…

Let R be a reflexive relation on a finite set A, and let MR be the bit matrix representing R. Specify the value of the entries on the main diagonal.

Let R be a symmetric relation on a finite set A, and let MR be the bit matrix representing R. Is MR necessarily a symmetric matrix? Why or why not?

Use Mathematical Induction to show that if MR is the bit matrix representing the relation R, then M^[n]R is the matrix representing R^n. (This was how the question was stated. If you're confused about the terms M^[n]R and R^n, they ar…

Let S be the set of bit strings of length no larger than 6, and define an equivalence relation R on S as follows: (x, y) ϵ R if and only if x and y are of the same length. Specify the partition P of S that arises from R.

Let S ={a, b, c, d, e}, and P={{a, b},{c, d},{e}}. (a) Verify that P really is a partiton of S. (b) Find the equivalence relation R on S induced by P.

Suppose S is a set, and P={S} is a partition of S. Find the equivalence relation R corresponding to P.

Let p and q be distinct positive primes. Define the partial ordering ⪯ of S={1, p, p^2, p^3, p^4, q, q^2, q^3, q^4} as follows: a⪯b if and only if a|b. Draw the Hasse diagram for this partial ordering of S.

Let S={1,2,3,4}, and define a partial ordering of P(S) (the power set of S), by: A⪯B if and only if A⊆B. Is this partial ordering in fact a total ordering (chain)? Why or why not?

Suppose S is a set containing 5 elements, and that ⪯ is a total ordering of S. Draw the Hasse diagram for ⪯ (no need to label the vertices in your diagram).

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Question ID: mtid-5-stid-8-sqid-713-qpid-598