Let P(x, y) be the proposition x
2 = y, where x and y are in the universe of integers. Determine the truth value of each of the following propositions.
(a) ∃x P(6, x) (b) ∃x P(x, 6) (c) ∀x ∃y P(x, y) (d) ∃y ∀x P(x, y)
2. For the domain of students enrolled at Macquarie let d(x) be the statement "x is enrolled in DMTH137" and r(x, y) be the statement "x has replied to a post by y on iLearn".
Express each of these sentences using predicates, quantifiers, and logical connectives.
(a) There is a DMTH137 student who replied to Jin on iLearn concerning COMP115 week 3 material.
(b) Every DMTH137 student has replied to a post to someone on iLearn.
(c) Nobody has responded to Deb's iLearn question about the due date for assignment 2 for DMTH137.
3. Prove or disprove each of the following propositions.
(a) If n 2 is a multiple of 4, then n is a multiple of 4.
(b) If n 3 is a multiple of 2, then n is a multiple of 2.
4. A number r is said to be rational if there exist integers a and b such that r = a/b.
An integer is a "counting number" such as 1, 2, 3, 4 and so on, or 0, or the negative of a counting number. Prove or disprove the following statements. You may assume that √
2 is irrational(a) The sum of rational numbers may be rational or irrational.
(b) The sum of two irrational numbers is irrational.
(c) The sum of a rational number and an irrational number is irrational also.
5. For this question, let the universal set U be the set of all people; let M be the set of males; let C be the set of all children, and let H be the set of all Dutch people. Express each of the following as sets:
(a) boys; (b) girls; (c) adult women; (d) adult Dutch men; (e) non-Dutch girls.
6. Show that A \ (B ∩ C) = (A \ B) ∪ (A \ C) by each of the following methods:
(a) Venn diagrams; (b) a subset argument; (c) a membership table.