Problem 1RQ: a) Define the negation of a proposition. b) What is the negation of "This is a boring course"? Problem 2RQ: a) Define (using truth tables) the disjunction, conjunction, exclusive or, conditional, and... Problem 3RQ: a) Describe at least five different ways to the conditional statementpq in English. b) Define the... Problem 4RQ: a) What does it mean for two propositions to be logically equivalent? b) Describe the different ways... Problem 5RQ: (Depends on the Exercise Set inSection 1.3) a) Given a truth table, explain how to use disjunctive... Problem 6RQ: What are the universal and existential quantifications of a predicateP(x)? What are their negations? Problem 7RQ: a) What is the difference between the quantificationxyP(x,y) andyxP(x,y) , whereP(x,y) is a... Problem 8RQ: Describe what is meant by a valid argument in propositional logic and show that the argument "If the... Problem 9RQ Problem 10RQ: a) Describe what is meant by a direct proof, a proof by contraposition, and a proof by contradiction... Problem 11RQ: a) Describe away to prove the bi-conditionalpq . b) Prove the statement: "The integer3n+2 is odd if... Problem 12RQ: To prove that the statementp1,p2,p3, andp4are equivalent, is it sufficient to show that the... Problem 13RQ: a) Suppose that a statement of the formxP(x) is false. How can this be proved? b) Show that the... Problem 14RQ: What is the difference between a constructive and non-constructive existence proof? Give an example... Problem 15RQ: What are the elements of a proof that there is a unique elementxsuch thatP(x), whereP(x) is a... Problem 16RQ Problem 1SE: Letpbe the proposition "I do every exercise in this book" andqbe the proposition "I will get an A in... Problem 2SE: Find the truth table of the compound proposition(pq)(pr) . Problem 3SE Problem 4SE Problem 5SE Problem 6SE Problem 7SE Problem 8SE Problem 9SE: Show that these statements are inconsistent: "If Miranda does not take a course in discrete... Problem 10SE: Suppose that in a three-round obligato game, the teacher first gives the student the propositionpq ,... Problem 11SE: Suppose that in a four-round obligato game, the teacher first gives the student the... Problem 12SE: Explain why every obligato game has a winning strategy. Exercises 13 and 14 are set on the island of... Problem 13SE Problem 14SE: Suppose that you meet three people, Anita, Boris, and Carmen. What are Anita, Boris, and Carmen if... Problem 15SE: (Adapted from [Sm78]) Suppose that on an island there are three types of people, knaves, and normals... Problem 16SE: Show that ifSis a proposition, whereSis the conditional statement "IfSis true, then unicorns live,"... Problem 17SE: Show that the argument premises "The tooth fry is a real person" and "The tooth fairy is not a real... Problem 18SE: Suppose that the truth value of the propositionPiis T wheneveriis an odd positive integer and is F... Problem 19SE: Model1616 Sudoku puzzles (with44 blocks) as satisfiability problems. Problem 20SE: Let P(x) be the statement “Student x knows calculus” and letQ(y) be the statement “Classycontains a... Problem 21SE: LetP(m,n) be the statement “mdividesn," where the domain for both variables consists of all positive... Problem 22SE: Find a domain for the quantifiers in xy(xyz((z=x)(z=y))) such that this statement is true. Problem 23SE Problem 24SE Problem 25SE: Use existential and universal quantifiers to express the statement "Everyone has exactly two... Problem 26SE: The quantifiern denotes "there exists exactlyn," so thatnxP(x) means there exist exactlynvalues in... Problem 27SE: Express each of these statements using existential and universal quantifiers and propositional... Problem 28SE Problem 29SE Problem 30SE: IfyxP(x,y) is true, does it necessarily follow that3xyP(x,y) is true? Problem 31SE Problem 32SE: Find the negations of these statements. a) If it snows today, then I will go skiing tomorrow. b)... Problem 33SE: Express this statement using quantifiers: "Every student in class has taken some course in every... Problem 34SE: Express statement using quantifiers: "There is a building on the campus of some college in the... Problem 35SE Problem 36SE Problem 37SE Problem 38SE: Prove that ifx3is irrational, thenxis irrational. Problem 39SE Problem 40SE Problem 41SE: Prove that there exists an integermsuch thatm2101000 . Is your proof constructive or... Problem 42SE Problem 43SE: Disprove the statement that every positive integer is the sum of the cubes of eight nonnegative... Problem 44SE: Disprove the statement that every positive integer is the sum of at most two squares and a cube of... Problem 45SE Problem 46SE: Assuming the truth of the theorem that states thatn is irrational whenevernis a positive integer... Problem 1CP: Given the truth values of the propositionspandq, find the truth values of the conjunction,... Problem 2CP Problem 3CP Problem 4CP Problem 5CP Problem 6CP: Given a portion of a checkerboard, look for tilings of this checkerboard with various types of... Problem 1CAE: Look for positive integers that are not the sum of the cubes of nine different positive integers. Problem 2CAE: Look for positive integers greater than 79 that are not the sum of the fourth powers of 18 positive... Problem 3CAE Problem 4CAE: Try to find winning strategies for the game of Chomp for different initial configurations of... Problem 5CAE Problem 6CAE: Find all the rectangles of 60 squares that can be tiled using every one of the 12 different... Problem 1WP: Discuss logical paradoxes, including the paradox of Epimenides the Cretan, Jourdain's card paradox,... Problem 2WP: Describe how fuzzy logic is being applied to practical applications. Consult one or more of the... Problem 3WP: Describe some of the practical problems that can be modeled as satisfiability problems. Problem 4WP Problem 5WP: Describe some of the techniques that have been devised to help people solve Sudoku puzzles the use... Problem 6WP: Describe the basic rules ofWFFN PROOF, The Game of Modern Logic, developed by Layman Allen. Give... Problem 7WP: Read some of the writings of Lewis Carroll on symbolic logic. Describe in detail some of the models... Problem 8WP: Extend the discussion of Prolog given inSection 1.4, explaining in more depth how Prolog employs... Problem 9WP: Discuss some of the techniques used in computational logic, including Skolem's rule. Problem 10WP: "Automated theorem proving" is the task of using computers to mechanically prove theorems. Discuss... Problem 11WP: Describe how DNA computing has been used to solve instances of the satisfiability problem. Problem 12WP: Look up some of the incorrect proofs of famous open questions and open questions that were solved... Problem 13WP Problem 14WP: Describe various aspects of proof strategy discussed by George Polya in his writings on reasoning,... Problem 15WP: Describe a few problems and results about tilings with polyominoes, as described in [Go94] and... format_list_bulleted