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Questions and Answers of Statistics

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. a) ∀x(x2 ≥ x) b) ∀x(x > 0 ∨x < 0) c) ∀x(x = 1)
Let T (x, y) mean that student x likes cuisine y, where the domain for x consists of all students at your school and the domain for y consists of all cuisines. Express each of these statements by a
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. a) ∃x∀y(xy = y) b) ∀x∀y(((x
Suppose the domain of the propositional functionP(x, y) consists of pairs x and y, where x is 1, 2, or 3 and y is 1, 2, or 3. Write out these propositions using disjunctions and conjunctions.a)
Find a common domain for the variables x, y, z, and w for which the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is true and another common domain for these variables for which
Let W(x, y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements
Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. a) ∀x∀y(x2 = y2 → x = y) b) ∀x∃y(y2 = x) c)
Determine the truth value of the statement ∀x∃y (xy = 1) if the domain for the variables consists of a) The nonzero real numbers. b) The nonzero integers. c) The positive real numbers.
Use quantifiers and logical connectives to express the fact that every linear polynomial (that is, polynomial of degree 1) with real coefficients and where the coefficient of x is nonzero, has
Find the argument form for the following argument and determine whether it is valid. Can we conclude that the conclusion is true if the premises are true? If Socrates is human, then Socrates is
Show that the argument form with premises p1, p2, . . . , Pn and conclusion q → r is valid if the argument form with premises p1, p2, . . . , Pn, q, and conclusion r is valid.
For each of these arguments, explain which rules of inference are used for each step.a) "Doug, a student in this class, knows how to write programs in JAVA. Everyone who knows how to write programs
For each of these arguments determine whether the argument is correct or incorrect and explain why. a) All students in this class understand logic. Xavier is a student in this class. Therefore,
What is wrong with this argument? Let H(x) be "x is happy." Given the premise ∃xH(x), we conclude that H(Lola). Therefore, Lola is happy.
Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? a) If n is a real number such that n > 1,
Which rules of inference are used to establish the conclusion of Lewis Carroll's argument described in Example 26 of Section 1.4?
Justify the rule of universal modus tollens by showing that the premises ∀x(P(x) → Q(x)) and ¬Q(a) for a particular element a in the domain, imply ¬P(a).
Use rules of inference to show that if ∀x(P(x) → (Q(x) ∧ S(x))) and ∀x(P(x) ∧ R(x)) are true, hen ∀x(R(x) ∧ S(x)) is true.
Use rules of inference to show that if ∀x(P(x) ∨ Q(x)), ∀x(¬Q(x) ∨ S(x)), ∀x(R(x)→¬S(x)), and ∃x¬P(x) are true, then ∃x¬R(x) is true.
What rule of inference is used in each of these arguments?a) Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major.b) Jerry is a mathematics major
Use resolution to show that the hypotheses "It is not raining or Yvette has her umbrella," "Yvette does not have her umbrella or she does not get wet," and "It is raining or Yvette does not get wet"
Use resolution to show that the compound proposition (p ∨ q) ∧ (¬p ∨ q) ∧ (p ∨¬q) ∧ (¬p ∨¬q) is not satisfiable.
Determine whether this argument, taken from Kalish and Montague [KaMo64], is valid. If Superman were able and willing to prevent evil, he would do so. If Superman were unable to prevent evil, he
Use rules of inference to show that the hypotheses "Randy works hard," "If Randy works hard, then he is a dull boy," and "If Randy is a dull boy, then he will not get the job" imply the conclusion
What rules of inference are used in this famous argument? "All men are mortal. Socrates is a man. Therefore, Socrates is mortal."
For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) "If I take the day
Use a direct proof to show that the sum of two odd integers is even.
Prove or disprove that the product of two irrational numbers is irrational.
Prove that if x is irrational, then 1/x is irrational.
Use a proof by contraposition to show that if x + y ≥ 2, where x and y are real numbers, then x ≥ 1 or y ≥ 1.
Show that if n is an integer and n3 + 5 is odd, then n is even using a) A proof by contraposition. b) A proof by contradiction.
Prove the proposition P(0), where P(n) is the proposition "If n is a positive integer greater than 1, then n2 > n." What kind of proof did you use?
Let P(n) be the proposition "If a and b are positive real numbers, then (a + b)n ≥ an + bn." Prove that P(1) is true. What kind of proof did you use?
Show that at least ten of any 64 days chosen must fall on the same day of the week.
Use a proof by contradiction to show that there is no rational number r for which r3 + r + 1 = 0. Assume that r = a/b is a root, where a and b are integers and a/b is in lowest terms. Obtain an
Prove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd.
Prove or disprove that if m and n are integers such that mn = 1, then either m = 1 and n = 1, or else m = −1 and n = −1.
Show that the square of an even number is an even number using a direct proof.
Show that these statements about the integer x are equivalent: (i) 3x + 2 is even, (ii) x + 5 is odd, (iii) x2 is even.
Show that these statements about the real number x are equivalent: (i) x is irrational, (ii) 3x + 2 is irrational, (iii) x/2 is irrational.
Are these steps for finding the solutions of √x + 3 = 3 − x correct? (1) √x + 3 = 3 − x is given; (2) x + 3 = x2 − 6x + 9, obtained by squaring both sides of (1); (3) 0 = x2 − 7x + 6,
Show that the propositions p1, p2, p3, p4, and p5 can be shown to be equivalent by proving that the conditional statements p1 → p4, p3 → p1, p4 → p2, p2 → p5, and p5 → p3 are true.
Prove that at least one of the real numbers a1, a2, . . . , an is greater than or equal to the average of these numbers. What kind of proof did you use?
Prove that if n is an integer, these four statements are equivalent: (i) n is even, (ii) n + 1 is odd, (iii) 3n + 1 is odd, (iv) 3n is even.
Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind of proof did you use?
Use a direct proof to show that every odd integer is the difference of two squares.
Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational.
Prove that n2 + 1 ≥ 2n when n is a positive integer with 1 ≤ n ≤ 4.
Prove that there exists a pair of consecutive integers such that one of these integers is a perfect square and the other is a perfect cube.
Prove or disprove that there is a rational number x and an irrational number y such that xy is irrational.
Show that each of these statements can be used to express the fact that there is a unique element x such that P(x) is true. a) ∃x∀y(P(y) ↔ x = y) b) ∃xP(x) ∧ ∀x∀y(P(x) ∧ P(y) → x =
Suppose that a and b are odd integers with a ≠ b. Show there is a unique integer c suc that |a - c| = |b - c|.
Show that if n is an odd integer, then there is a unique integer k such that n is the sum of k − 2 and k + 3.
Prove that given a real number x there exist unique numbers n and ε such that x = n - ε, n is an integer, and 0 ≤ ε < 1.
The harmonic mean of two real numbers x and y equals 2xy/(x + y).By computing the harmonic and geometric means of different pairs of positive real numbers, formulate a conjecture about their relative
Write the numbers 1, 2, . . . , 2n on a blackboard, where n is an odd integer. Pick any two of the numbers, j and k, write |j − k| on the board and erase j and k. Continue this process until only
Formulate a conjecture about the decimal digits that appear as the final decimal digit of the fourth power of an integer∙ Prove your conjecture using a proof by cases.
Prove that there is no positive integer n such that n2 + n3 = 100.
Prove that if x and y are real numbers, then max(x, y) + min(x, y) = x + y. Use a proof by cases, with the two cases corresponding to x ≥ y and x < y, respectively.
Prove that there are no solutions in positive integers x and y to the equation x4 + y4 = 625.
Adapt the proof in Example 4 in Section 1.7 to prove that if n = abc, where a, b, and c are positive integers, thena ≤ 3√n, b ≤ 3√n, or c ≤ 3√n.
Prove that between every two rational numbers there is an irrational number.
S = x1y1 + x2y2 +· · · + xnyn, where x1, x2, . . . , xn and y1, y2, . . . , yn are orderings of two different sequences of positive real numbers, each containing n elements. a) Show that S takes
Verify the 3x + 1 conjecture for these integers. a) 6 b) 7 c) 17 d) 21
Prove or disprove that you can use dominoes to tile the standard checkerboard with two adjacent corners removed (that is, corners that are not opposite).
Prove that you can use dominoes to tile a rectangular checkerboard with an even number of squares.
Use a proof by exhaustion to show that a tiling using dominoes of a 4 × 4 checkerboard with opposite corners removed does not exist. First show that you can assume that the squares in the upper left
Show that by removing two white squares and two black squares from an 8 × 8 checkerboard (colored as in the text) you can make it impossible to tile the remaining squares using dominoes.
a) Draw each of the five different tetrominoes, where a tetromino is a polyomino consisting of four squares. b) For each of the five different tetrominoes, prove or disprove that you can tile a
Prove using the notion of without loss of generality that min(x, y) = (x + y − |x − y|)/2 and max(x, y) = (x + y + |x − y|)/2 whenever x and y are real numbers.
Prove the triangle inequality, which states that if x and y are real numbers, then |x| + |y| ≥ |x + y| (where |x| represents the absolute value of x, which equals x if x ≥ 0 and equals −x if x
Prove that there are 100 consecutive positive integers that are not perfect squares. Is your proof constructive or non-constructive?
Let p be the proposition "I will do every exercise in this book" and q be the proposition "I will get an "A" in this course." Express each of these as a combination of p and q.a) I will get an "A" in
Suppose that in a four-round obligato game, the teacher first gives the student the proposition ¬(p → (q ∧ r)), then the proposition p ∨¬q, then the proposition¬r, and finally, the
Suppose that you meet three people Aaron, Bohan, and Crystal. Can you determine what Aaron, Bohan, and Crystal are if Aaron says "All of us are knaves" and Bohan says "Exactly one of us is a knave."?
(Adapted from [Sm78]) Suppose that on an island there are three types of people, knights, knaves, and normals (also known as spies). Knights always tell the truth, knaves always lie, and normals
Show that the argument with premises "The tooth fairy is a real person" and "The tooth fairy is not a real person" and conclusion "You can find gold at the end of the rainbow" is a valid argument.
Model 16 × 16 Sudoku puzzles (with 4 × 4 blocks) as satisfiability problems.
Let P(m, n) be the statement "m divides n," where the domain for both variables consists of all positive integers. (By "m divides n" we mean that n = km for some integer k.) Determine the truth
Use existential and universal quantifiers to express the statement "Everyone has exactly two biological parents" using the propositional function P(x, y), which represents "x is the biological parent
Express each of these statements using existential and universal quantifiers and propositional logic where ∃n is defined in Exercise 26.a) ∃0xP(x)b) ∃1xP(x)c) ∃2xP(x)d) ∃3xP(x)
Let P(x) and Q(x) be propositional functions. Show that ∃x (P(x) → Q(x)) and ∀x P(x) → ∃x Q(x) always have the same truth value.
Show that these compound propositions are tautologies. a) (¬q ∧ (p → q)) → ¬p b) ((p ∨ q) ∧ ¬p) → q
If ∀x ∃y P(x, y) is true, does it necessarily follow that ∃x ∀y P(x, y) is true?
Express this statement using quantifiers: "Every student in this class has taken some course in every department in the school of mathematical sciences."
Express the statement "There is exactly one student in this class who has taken exactly one mathematics class at this school" using the uniqueness quantifier. Then express this statement using
Prove that if x is irrational and x ≥ 0, then √x is irrational.
Disprove the statement that every positive integer is the sum of the cubes of eight nonnegative integers.
Disprove the statement that every positive integer is the sum of 36 fifth powers of nonnegative integers.
Given a conditional statement p → q, find the converse of its inverse, the converse of its converse, and the converse of its contrapositive.
Find a compound proposition involving the propositional variables p, q, r, and s that is true when exactly three of these propositional variables are true and is false otherwise.
Show that these statements are inconsistent: "If Miranda does not take a course in discrete mathematics, then she will not graduate." "If Miranda does not graduate, then she is not qualified for the
Describe various aspects of proof strategy discussed by George Pólya in his writings on reasoning, including [Po62], [Po71], and [Po90].
Describe some of the practical problems that can be modeled as satisfiability problems.
Describe the basic rules of WFF'N PROOF, The Game of Modern Logic, developed by Layman Allen. Give examples of some of the games included in WFF'N PROOF.
Read some of the writings of Lewis Carroll on symbolic logic. Describe in detail some of the models he used to represent logical arguments and the rules of inference he used in these arguments.
Extend the discussion of Prolog given in Section 1.4, explaining in more depth how Prolog employs resolution.
Discuss some of the techniques used in computational logic, including Skolem's rule.
List the members of these sets. a) {x | x is a real number such that x2 = 1} b) {x | x is a positive integer less than 12} c) {x | x is the square of an integer and x < 100} d) {x | x is an integer
Determine whether each of these statements is true or false. a) x ∈ {x} b) {x} ⊆ {x} c) {x} ∈ {x} d) {x} ∈ {{x}} e) ∅ ⊆ {x} f) ∅ ∈ {x}

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