Solved: Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.] a) It is necessary to wash the boss’s car to get promoted. b) Winds from the south imply a spring thaw. c) A sufficient condition for the warranty to be good is that you bought the computer less than a year ago. d) Willy gets caught whenever he cheats. e) You can access the website only if you pay a subscription fee. f ) Getting elected follows from knowing the right people. g) Carol gets seasick whenever she is on a boat.

Let f : A → B be a function and σ an equivalence relation on B. Define a relation ρ on A as: a ρ a' if and only if f(a) σ f(a'). 1. Prove that ρ is an equivalence relation on A. 2. Define a map f : A/ρ → B/σ as [a]ρ 7→ [f(a)]σ. Prov…

1) A person deposits 1000 USD into an account that yields 9 percent interest compounded annually. Let An denote the amount of money in the account after n years. (a) Set up a recurrence relation for An. (b) Find an explicit formula fo…

Suppose that the number of bacteria in a colony triples every hour. Let Bn denote the number of bacteria in the colony after n hours. (a) Set up a recurrence relation for Bn. (b) If 100 bacteria are used to begin a new colony, how man…

Prove or disprove that there exists a bijection from (0, 1] to (0, 1]^2

Prove or disprove that there exists a bijection from (0, 1] to [0, ∞)^2.

Let ρ be a relation on a set A. Define ρ^−1 = {(b, a) | (a, b) ∈ ρ}. Also, for two relations ρ, σ on A, define the composite relation ρ ◦ σ as (a, c) ∈ ρ ◦ σ if and only if there exists b ∈ A such that (a, b) ∈ ρ and (b, c) ∈ σ. Prove…

In class we showed the following: n ∑(k=1) k = n(n+1)/2 and n∑(k=1) k^2 = n(n+ 1)(2n+ 1)/6 Using the fact that (k+ 1)^4-k^4= 4k^3+ 6k^2+ 4k+ 1 and summing up as k= 1,2,3,……, n together with the above two equalities, deduce that n∑(k=1…

Compute the values of the sums below. (a)5∑(k=1) (k+ 1) (b)4∑(j=0) (-2) ^j (c) 10∑(i=1) 3 (d) 8∑(j=0) ( 2^(j+1) -2^j )

What are the values of these sums, where S={1,3,5,7}? (a)∑(jϵS) j (b)∑(jϵS) j^2 (c)∑(jϵS) (1/j) (d)∑(jϵS) 1

Compute each of the double double sums below (a)3∑(i=1) 2∑(j=2) (i-j) (b)3∑(i=0) 2∑(j=0) (3i+2j) (c)3∑(i=1) 2∑(j=0) j (d) 2∑(i=0) 3∑(j=0) i^2 j^3

(7) An employee joined a company in 2017 with a starting salary of 50000 USD. Every year this employee receives a raise of 1000 USD plus 5 percent of the salary of the previous year. Let Cn denote the employee's salary n years after 2…

Show that a subset of a countable set is also countable

Show that if A and B are sets where A⊆B and A is uncountable, then B is uncountable.

If A is an uncountable set and B is a countable set, must A-B be uncountable?

Show that if A, B, C, and D are sets with |A|=|B| and|C|=|D|, then |AxC|=|BxD|.

Let A be a set, and let P(A) denote the power set of A. Prove that|A|<|P(A)|. Hint: Proceed in two steps. 1. First show that|A| <= |P(A)|. Try defining the function g: A-> P(A) by g(a) ={a}, and verify that g is one-to-one. 2. Then sh…

What time does a 24-hour clock read (a) 100 hours after it reads 2:00? (b) 45 hours before it reads 12:00? (c) 168 hours after it reads 19:00?

Suppose a and b are integers, a ≡ 11 (mod 19) and b ≡ 3 (mod 19). Find an integer c with 0≤c≤18 such that (a)c≡13a(mod 19). (b)c≡8b(mod 19). (c)c≡a-b(mod 19). (d)c≡7a+ 3b(mod 19). (e)c≡2a^2+ 3b^2(mod 19). (f)c≡a^3+ 4b^3(mod 19).

Decide whether each of these integers is congruent to 3 modulo 7. (a) 37 (b) 66 (c) -17 (d) -67

Show that if a≡b(mod m) and c≡d(mod m), where a, b, c, d ϵ Z with m≥2, then a-c≡b-d(mod m).

Show that if n|m, where n and m are integers greater than 1, and if a≡b(mod m), where a and b are integers, then a≡b(mod n).

(19) Show that if n is an integer then n^2 ≡ 0 or 1 (mod 4). (20) Use the result of Exercise (19) to show that if m is a positive integer that can be written in the form m= 4k+ 3 (where k is a nonnegative integer), then m is not the s…

Prove that if n is an odd positive integer, then n^2 ≡ 1 (mod 8).

Show that if A and B are sets with the same cardinality, then |A|<=|B| and |B|<=|A|

The recursive definition of a function X is given as: f(0)=5 and f(n)=f(n-2)+5 Now, find out the value of f(14) using the above function.

Explain graph isomorphism with proper diagrams.

Define the following terms with proper example: a. Relatively prime integers b. Hadamard Matrix

Define the following terms with proper diagrams: a. Degree of a node in a graph b. Copies of binary trees

Instruction: Write only the letter of your answer. 1. Suppose p is the statement 'You need a credit card' and q is the statement 'I have a nickel.' Select the correct statement corresponding to the symbols ┐(p ∨ q). A. You don't need …

4. Suppose p is the statement 'I play softball' and q is the statement 'The moon is 250,000 miles from Earth.' Select the correct statement corresponding to the symbols ┐p ∧ q. A. I don't play softball and the moon is 250,000 miles fr…

7-13: Find the Truth Value (show your solution) 7. Suppose p is false, q is false, s is true. Find the truth value of (s ∨ p) ∧ (q∧ ┐s) 8. Suppose p is true, q is true, r is false, s is false. Find the truth value of (s ∨ p) ∧ (┐r ∨ ┐…

6. Select the statement that is the negation of “You wear matching socks to the interview or you don’t get hired.” A. You don’t wear matching socks to the interview or you get hired. B. You don’t wear matching socks to the interview a…

Explain graph isomorphism with proper diagrams.

The recursive definition of a function X is given as: f(0)=5 and f(n)=f(n-2)+5 Now, find out the value of f(14) using the above function.

Consider the relation on the set of integer R={(a, b) /a=b+1} check whether it is equivalence relation

Among the integer 1 to 300,find how many are not divisible by 3,nor by 5 also find how many are divisible by 3but not by 7

The recursive definition of a function X is given as: f(0)=5 and f(n)=f(n-2)+5 Now, find out the value of f(14) using the above function.

How many bits would be required to represent decimal numbers from -32,768 to +32,767?

Each of the following numbers represents a signed decimal number in the 2’s complement system. Determine the decimal value in each case  10011001  11101  01111011

DISCUSS AND ILLUSTRATE WITH EXAMPLES. 15 POINTS EACH QUESTION. B. Universal Relations C. Identity Relations D. Inverse Relations E. Reflexive Relations F. Symmetric Relations G. Transitive Relations

For the function f defined by f(n) =n2+ 1/n+ 1 for n∈N, show that f(n)∈Θ(n). Use

If n is any even integer and m is any odd integer, then (n + 2)2 - (m - 1)2 is even.

Suppose a ϵ Z. Prove by contradiction that if a2 -2a + 7 is even, then a is even.

Use proof by contraposition to show that if x + y ≥ 2, where x and y are real numbers, then x ≥ 1 or y ≥ 1.

Suppose a ϵ Z. Prove by contradiction that if a2 -2a + 7 is even, then a is even.

Suppose that in the world every pair of people either (a) likes one another, (b) dislikes one another, or (c) is indifferent toward one another. Prove that in any gathering of 17 people, there is a group of three people all of whom sa…

In a a class of 80 students 53 study Art 60 study Biology 36 study art and Biology 34 study art and Chemistry 6 study Biology only and 18 study biology but not chemistry. How many students study art only and how many students study ch…

. The crossing number of a graph G, written ν(G), is the fewest number of nonendpoint edge-crossings that occur over all possible drawings of G in the plane. We assume that no edge crosses itself, and that edge crossings occur only at…

WHAT IS THE CARDINALITY OF {a,{a},{a,{a}}}

Use Karnaugh map to minimize the sum of product expansion xy'z+ xy'z'+x'yz+x'y'z+x'y'z'

Determine whether f is a function from the integers to the set of all real numbers. Enter "Y" for yes and "N" for no. 1. f(n)=±n 2. f(n)=1/(n^2−16) 3. f(n)=√n^2+6 4. f(n)=1/(n^2+9)

Find the following values. (a) ⌊1.1⌋ (b) ⌈1.1⌉ (c) ⌊−0.1⌋ (d) ⌈−0.1⌉ (e) ⌈2.99⌉ (f) ⌈−2.99⌉ (g) ⌊1/2+⌈1/2⌉⌋ (h) ⌈⌊1/2⌋+⌈1/2⌉+1/2⌉

Determine if each of the following functions from {a,b,c,d} to itself is one-to-one and/or onto. Check ALL correct answers. (a) f(a)=d,f(b)=a,f(c)=c,f(d)=b A. onto. B. neither one-to-one nor onto. C. one-to-one. f(a)=b,f(b)=a,f(c)=c…

Given that f(x)=8x^2+7 and g(x)=4x+4 are functions from R to R, find (a) f∘g. (b) g∘f.

(a) How many bit strings of length 8 are there? (b) How many bit strings of length 8 or less are there? (Count the empty string of length zero also.) (c) How many strings of 6 lower case English letters are there that have the letter …

Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties: (a) have distinct digits. (b) are divisible by 5 and by 7. (c) are even. (d) …

How many strings of four decimal digits (Note there are 10 possible digits and a string can be of the form 0014 etc., i.e., can start with zeros.) (a) do not contain the same digit twice? (b) begin and end with a 1?

How many strings of five uppercase English letters are there (a) that start or end with the letters BO (in the order), if letters can be repeated? (inclusive or) (b) that start with the letters BO (in that order), if letters can be re…

Solve the following two " union " type questions: (a) How many bit strings of length 8 either begin with 2 0s or end with 1 1s? (inclusive or) (b) Every student in a discrete math class is either a computer science or a math…

bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. (a) How many balls must she select (minimum) to be sure of having at least three blue balls? (b) How many balls must she select (mi…

This question concerns bit strings of length six. These bit strings can be divided up into four types depending on their initial and terminal bit. Thus the types are: 0XXXX0, 0XXXX1, 1XXXX0, 1XXXX1. How many bit strings of length six …

Find the value of each of the following quantities: C(8,4)= C(12,7)= C(12,8)= C(11,1)= C(10,9)= C(8,6)=

How many bit strings of length 14 have: (a) Exactly three 0s? (b) The same number of 0s as 1s? (d) At least three 1s?

15 players for a softball team show up for a game: (a) How many ways are there to choose 10 players to take the field? 10!/(5!) (b) How many ways are there to assign the 10 positions by selecting players from the 15 people who show…

Suppose that a department contains 8 men and 20 women. How many ways are there to form a committee with 6 members if it must have strictly more women than men?

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?

Find the coefficient of x^5 in (1+x)^11.

What is the coefficient of x^3 y^13 in the expansion of (−1x+2y)^16?

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?

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Question ID: mtid-5-stid-8-sqid-644-qpid-529