Solved: Translate the following statement into mathematical expression based on the given propositional variables. “If a country’s leaders are elected based on good leadership traits, then the country will have good governance”. p : A country’s leaders are elected based on good leadership traits. q : The country will have a good governance. q → pp → qp ⟷ qq ⟷ p

Show that these three statements are equivalent, where a and b are real numbers: (i)a < b (ii) The average of a and b is greater than a. (iii) The average of a and b is less than b.

Show that these statements about the real number x are equivalent: (i)x is rational. (ii)x/2 is rational. (iii) 3x-1 is rational.

3) Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers.

14) Show that if the first 10 positive integers are placed around a circle, in any order, then there exist three integers in consecutive locations around the circle that have a sum of at least 17

Show that 5x+ 5y is an odd integer when x and y are integers of opposite parity.

Show that if a, b and c are real numbers and a=/= 0, then there is a unique solution of the equation ax+b=c.

Show that if r is an irrational number, there is a unique integer n such that the distance between r and n is less than 1/2

Prove that there are no solutions in integers x and y to the equation 2x^2+ 5y^2= 14.

Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x^2+y^2=z^2. Hint: Let x=m^2-n^2 ,y= 2mn, and z = m^2+n^2, where m and n are integers.

Prove that there is a positive integer that equals the sum of the positive integersnot exceeding it

Simplify (a’ * b’ * c) + (a * b’ * c) + (a * b’ * c’)

In the application for admission to the master's degree of 350 students. Suppose that 220 of these applicants majored in computer science, 147 majored in business, and 51 majored both in computer science and in business. How man…

Prove that f:R→ R defined by f(x)=x3 -5 is a bijection

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

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

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?

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