Solved: How many numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no repetition is allowed? 13) Suppose that we draw a card from a deck of 52 cards and replace it before the next draw. In how many ways can 10 cards be drawn so that the tenth card is a repetition of a previous draw?

Show that ∀xP(x)∧∃xQ(x) is logically equivalent to ∀x∃y(P(x)∧Q(y)), where all quantifiers have the same nonempty domain.

1. Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that A= {l, 3, 4, 8}, B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit representations for A, B, and C together with UNION, INTER, DIFF, and…

Determine the truth value of each of ∀x∀y∃z(z=(x+y)/2)

(4) Find a common domain for the variables x, y, and z for which the statement ∀x∀y((x=/=y) => ∀z((z=x)v(z=y))) is true and another domain for which it is false.

(5) Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. (a)∀x∃y(x= 1/y) (b)∀x∃y(y^2-x <100) (c)∀x∀y(x^2=/=y^3)

(6) Determine the truth value of the statement ∃x∀y(x lessthanorequalto y2) if the domain for the variables consists of (a) The positive real numbers. (b) The integers. (c) The nonzero real numbers.

(7) The Logic Problem has two assumptions: (i) "Logic is difficult or not many students like logic." (ii) "If mathematics is easy, then logic is not difficult." By translating these assumptions into statements invo…

(8) Determine whether the argument below is valid: "If Superman were able and willing to prevent evil, he would do so. If Superman were unable to prevent evil, he would be impotent; if he were unwilling to prevent evil, he would …

Prove that 3Z is an infinite set

Give an example to prove that intersection of denumerable sets need not be denumerable

(1) Let Q(x, y) be the statement "x+y=x-y." If the domain for both variables consists of all integers, what are the truth values? (a)Q(1,1) (b)Q(2,0) (c)∀yQ(1, y) (d)∃xQ(x,2) (e)∃x∃yQ(x, y) (f)∀x∃yQ(x, y) (g)∃y∀xQ(x, y) (h)∀…

Prove that 3Z is infinite

Let X = N (natural numbers) - {0} and Y= P(X) - {fi} f: X -> Y g: Y-> Y h: Y-> X f(x) = {n belongs to X: n is a divisor of x}, for all x belongs to X g(A)= {n belongs to X: n-2 belongs to A}, for all A belongs to Y h(A) = t…

in 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 bot chemistry. illustrate the information on a venn diagram

Use a direct proof to show that the sum of two even integers is even

(2) Show that the additive inverse, or negative, of an even number is an even num-ber using a direct proof

Use a direct proof to show that the product of two odd numbers is odd.

Prove that if n is a perfect square, then n+ 2 is not a perfect square.

Use a direct proof to show that the product of two rational numbers is rational.

Prove or disprove that the product of a nonzero rational number and an irrational number is irrational

Prove that if x is rational and x=/= 0, then 1/x is rational.

Prove that if n is an integer and 3n+ 2 is even, then n is even.

(9) Show that at least three of any 25 days chosen must fall in the same month of the year

Prove that if n is a positive integer, then n is even if and only if 7n+ 4 is even.

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}}}

How many numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no repetition is allowed? 13) Suppose that we draw a card from a deck of 52 cards and replace it before the next draw. In how many ways can 10 cards be drawn so that the tenth card is a repetition of a previous draw?

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