Solved: In a school, 100 students have access to three software packages A, B and C. 28 did not use any software 8 used only packages A 26 used only packages B 7 used only packages C 10 used all three packages 13 used both A and B If twice as many students used package B as package A, Using Venn Diagram: 1. Calculate the no. of students using packages A & C only but not B. 2. Calculate the no. of students using packages C & B only but not A.

Draw the graph of the function f(n) = 3n + 1 from the set of integers to the set of integers.

Draw the graph of the function f(x) = 3x 2 + 1 from the set of real numbers to the set of real numbers.

Determine whether each of these functions from [a,b,c,d] to itself is one-to-one. (a) f(a) = b, f(b) = a, f(c) = c, f(d) =d (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c

Let P(x) be the statement “x likes subject mathematics,” where the domain for x consists of all students. Express each of these quantifications in English. a) ∃

5. Let C(x, y) be the statement “x is a friend of y,” where the domain for x and y consists of all people. Use quantifications to express each of the following statements. a) Everyone is a friend of everyone. b) Not everyone is a frie…

An explorer is captured by a group of cannibals. There are two types of cannibals: those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular canniba…

1. Build up the operation tables for group G with orders 1, 2, 3 and 4 using the elements a, b, c, and e as the identity element in an appropriate way. 2. i. State the Lagrange’s theorem of group theory. ii. For a subgroup H of a grou…

The following exercises relate to inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth and knaves always lie. You encounter two people,A and B.Determine, if possible, what A and B ar…

The following exercises relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth,knaves who always lie, and spies who can either lie or tell the truth. You encounter three people,A…

A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has concluded that if the butler is telling the truth then so is the cook; the cook and the gardener cannot both be telling the tru…

For the given sequence 2, 6, 10, ... i- Write down the recursive formula for above sequence ii- Write down the closed formula for above sequence iii- Find the 11th term of the given sequence.

(1) Let Q(x, y) be the statement "x+y=xy." 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)∀y…

(2) Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. (a)∀x∃y(x^2=y) (b)∀x∃y(x=y^2) (c)∃x∀y(xy= 0) (d)∃x∃y(x+y=/=y+x) (e)∀x(x=/= 0 => ∃y(xy= 1)) (f)∃x∀y(y=/= 0 => xy= 1)…

(3) Express the negations of each of these statements so that all negation symbols immediately precede predicates. (a)∃z∀y∀xT(x, y, z) (b)∃x∃yP(x, y)^∀x∀yQ(x, y) (c)∃x∃y(Q(x, y)<=>Q(y,x)) (d)∀y∃x∃z(T(x, y, z)∨Q(x, y))

Show that (p∧q=⇒ r and (p=⇒r)∧(q=⇒ r) are not logically equivalent.

Suppose that a truth table in n propositional variables is specified. Show that a compound proposition with this truth table can be formed by taking the disjunction of conjunctions of the variables or their negations, with one conjunc…

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next,express the negation in simple English. (a) There is a student in this class who has …

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 ∨ ┐…

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