King saud logic

Submit Homework 1 | Gradescope 2/8/24, 6:04 PM 0/11 Questions Answered https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 1 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Homework 1 Q1 Instructions 1 Point Grading: Homework assignments will be graded traditionally with partial credits and you will be given individualized feedback. Excellent solutions take many forms, but they all have the following characteristics: 1. they use complete sentences, even when formulas or symbols are involved; 2. they are written as explanations for other students in the course; in particular, they fully explain all of their reasoning and do not assume that the reader will fill in details; 3. when graphical reasoning is called for, they include large, carefully drawn and labeled diagrams. Academic Honesty: You may discuss the homework problems with your classmates. Talking to others through your ideas, and suggesting techniques or procedures for others to try is encouraged! However, avoid showing other students your completed write-up or working on the problem for them. Everyone must write their solutions independently. I know that finding solutions on the Internet is not hard. While these might be useful tools for studying for exams, copying said solutions on a homework assignment will result in a negative grade for that assignment. Save Answer https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 2 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Q2 Translating English Sentence using connectives 1 Point Let p and q be the propositions p: You drive over 65 miles per hour. q : You get a speeding ticket. Write these propositions using p and q and logical connectives (including negations). a) You do not drive over 65 miles per hour. b) You drive over 65 miles per hour, but you do not get a speeding ticket. c) You will get a speeding ticket if you drive over 65 miles per hour. d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket. e) Driving over 65 miles per hour is sufficient for getting a speeding ticket. f) You get a speeding ticket, but you do not drive over 65 miles per hour. g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.  Please select file(s) Select file(s) Save Answer https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 3 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Q3 Converse, Contrapositive, and Inverse 1 Point State the converse, contrapositive, and inverse of each of these conditional statements. a) If it snows tonight, then I will stay at home. b) I go to the beach whenever it is a sunny summer day. c) When I stay up late, it is necessary that I sleep until noon.  Please select file(s) Select file(s) Save Answer Q4 Construct a truth table 1 Point Construct a truth table for each of these compound propositions. (p → q) ∧ (¬p → r)  Please select file(s) Select file(s) Save Answer https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 4 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Q5 De Morgan’s laws to find the negation 1 Point Use De Morgan’s laws to find the negation of each of the following statements. a) Kwame will take a job in industry or go to graduate school. b) James is young and strong.  Please select file(s) Select file(s) Save Answer Q6 Tautology 1 Point Show that each of these conditional statements is a tautology by using truth tables. a) (p ∧ q) → p b) p → (p ∨ q)  Please select file(s) Select file(s) Save Answer https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 5 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Q7 Logically equivalence 1 Point By using Equivalence Laws (not the truth table), show that (p → q) ∨ (p → r) ≡ p → (q ∨ r).  Please select file(s) Select file(s) Save Answer Q8 Deduction Rule 1 Point Determine if the following (P ∧ Q) → R ¬P ∨ ¬Q ∴ ¬R is a valid deduction rule.  Please select file(s) Select file(s) Save Answer https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 6 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Q9 Quantifiers 1 Point As mentioned in the text, the notation ∃! x P (x) denotes “There exists a unique x such that P (x) is true.” If the domain consists of all integers, what are the truth values of these statements? a) ∃!x(x > 1) b) ∃!x (x2 = 1) c) ∃!x(x + 3 = 2x) d) ∃!x(x = x + 1)  Please select file(s) Select file(s) Save Answer https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 7 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Q10 Nested Quantifiers 1 Point Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a) ∀x∃y (x2 = 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))  Please select file(s) Select file(s) Save Answer https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Page 8 of 9 Submit Homework 1 | Gradescope 2/8/24, 6:04 PM Q11 Negation of Quantifiers 1 Point Rewrite each of these statements so that negations appear only within predicates (that is so that no negation is outside a quantifier or an expression involving logical connectives). a) ¬(∀x∀yP (x, y)) b) ¬(∀y∃xP (x, y)) c) ¬(∀y∀x(P (x, y) ∨ Q(x, y)))  Please select file(s) Select file(s) Save Answer Save All Answers https://www.gradescope.com/courses/708781/assignments/4012950/submissions/new Submit & View Submission  Page 9 of 9