Discrete Mathematics 4 questions

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Discrete Mathematics (MAT 214) Homework 3 Complete the following problems on a separate sheet of paper. Write as much as possible for full credit, including calculator work. This assignment is due on Wednesday, February 6th . 1) Let Q(n) be the predicate “n2 ≤ 30.” a) Write Q(2), Q(−2), Q(7), and Q(−7), and indicate which of these statements are true and which are false. b) Find the truth set of Q(n) if the domain of n is Z, the set of all integers. c) If the domain is the set Z+ , the set of all positive integers, what is the truth set of Q(n). 2) Rewrite each of the following statements in the form “∀ a) Every real number is positive, negative, or zero. b) The number −1 is not equal to the square of any real number. , .” 3) Let D = {−48, −14, −8, 0, 1, 3, 16, 23, 26, 32, 36}. Determine which of the following statements are true and which are false. Provide counterexamples for those statements that are false. a) ∀x ∈ D, if x is odd then x > 0. b) ∀x ∈ D, if x is less than 0 then x is even. c) ∀x ∈ D, if x is even then x ≤ 0. d) ∀x ∈ D, if the ones digit of x is 2, then the tens digit is 3 or 4. e) ∀x ∈ D, if the ones digit of x is 6, then the tens digit is 1 or 2. 4) Write a formal negation for each of the following statements: a) ∀ fish x, x has gills. b) ∀ computers c, c has a CPU. c) ∃ a movie m such that m is over 6 hours long. d) ∃ a band b such that b has won at least 10 Grammy awards.
Discrete Mathematics (MAT 214) Valid Argument Forms The following is a list of common Valid Argument Forms. Given any assumptions (premises) in the given form, we can always conclude on the given conclusion. Modus Ponens p→q p ∴ q Modus Tollens p→q ∼q ∴ ∼p Generalization p ∴ p∨q q ∴ p∨q Specialization p∧q ∴ p p∧q ∴ q Conjunction p q ∴ p∧q Elimination p∨q ∼q ∴ p Transitivity p→q q→r ∴ p→r Proof by Division into Cases p∨q p→r q→r ∴ r Contradiction Rule ∼p → c ∴ p p∨q ∼p ∴ q

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School: Duke University

Hello, I am through with the work. Please find it in the attached document below. Am here to help in future when you have more work. Thanks and good luck in your education.



Discrete Mathematics
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1. Let Q(n) be th...

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Tutor went the extra mile to help me with this essay. Citations were a bit shaky but I appreciated how well he handled APA styles and how ok he was to change them even though I didnt specify. Got a B+ which is believable and acceptable.

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