Solved: j=0 ∑ 8 ( j 8 ) (j+1)(j+2) 1 =1⋅ (0+1)(0+2) 1 +8\cdot\dfrac{1}{(1+1)(1+2)}+28\cdot\dfrac{1}{(2+1)(2+2)}+8⋅ (1+1)(1+2) 1 +28⋅ (2+1)(2+2) 1 +56\cdot\dfrac{1}{(3+1)(3+2)}+70\cdot\dfrac{1}{(4+1)(4+2)}+56⋅ (3+1)(3+2) 1 +70⋅ (4+1)(4+2) 1 +56\cdot\dfrac{1}{(5+1)(5+2)}+28\cdot\dfrac{1}{(6+1)(6+2)}+56⋅ (5+1)(5+2) 1 +28⋅ (6+1)(6+2) 1 +8\cdot\dfrac{1}{(7+1)(7+2)}+1\cdot\dfrac{1}{(8+1)(8+2)}+8⋅ (7+1)(7+2) 1 +1⋅ (8+1)(8+2) 1 =\dfrac{1013}{90}= 90 1013

Determine whether the statements below are true or false. (a) x ϵ {x} (b) {x} ⊆ {x} (c) {x} ϵ {x} (d) {x} ϵ {{x}} (e) {Ø} ⊆ {Ø,,{Ø}} (f) Ø ⊆ {x} (g) Ø ⊆ {x}

7(pv(7p^q)) and 7p^7q are logically equivalence by developing a series of logical equivalence

show that 7(p->q) and p^7 q are logically equivalent without using truth table or using identities

Determine whether the statements below are true or false. (a) x ϵ {x} (b) {x} ⊆ {x} (c) {x} ϵ {x} (d) {x} ϵ {{x}} (e) {Ø} ⊆ {Ø,,{Ø}} (f) Ø ⊆ {x} (g) Ø ⊆ {x}

Question #135351 1. Let Q(x) denote the statement “x=x+1” What is the truth value of the quantification ∃x Q(x), where the domain consists of all real numbers?

There are 6 routes from Delhi to Mumbai and 12 routes from Mumbai to Bangalore. In how many ways can you travel from Delhi to Bangalore via Mumbai? 18 72 36 16

Let U = {English, French, History, Math, Physics, Chemistry, Psychology, Drama}, A = {English, Chemistry, French, Psychology}, B = {Math, Physics, History, French, Psychology}, and C = {Drama, Chemistry, History}. Find the following.…

Construct a proof for the five color theorem for every planar graph. 2. Discuss how efficiently Graph Theory can be used in a route planning project for a vacation trip from Colombo to Trincomalee by considering most of the practical …

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

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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}}}

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(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) …

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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…

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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?

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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

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

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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.

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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

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Question ID: mtid-5-stid-8-sqid-636-qpid-521