MATH1240 W23 Test 1 Solutions

UNIVERSITY OF MANITOBA COURSE: MATH 1240 DATE: February 2023 Test 1 DURATION: 50 minutes I understand that cheating is a serious offence: Signature : ( In Ink ) INSTRUCTIONS I. No texts, notes, or other aids are permitted. There are no calculators, cellphones or electronic translators permitted. II. This exam has a title page, 6 pages including this cover page. Please check that you have all the pages. Do not remove any pages. III. The value of each question is indicated in the lefthand margin beside the statement of the question. The total value of all questions is 28 points. IV. Answer all questions on the exam paper in the space provided beneath the question. Unjustified answers will receive little or no credit. If you need more space, continue on the back of the page, CLEARLY INDICATING THAT YOUR WORK IS TO BE CONTINUED.

UNIVERSITY OF MANITOBA COURSE: MATH 1240 Test 1 DATE & TIME: February 2023, DURATION: 50 minutes V. Do not deface the QR code in the top right corner. Doing so may result in the page not being scanned and therefore not graded.

UNIVERSITY OF MANITOBA COURSE: MATH 1240 DATE: February 2023 Test 1 DURATION: 50 minutes 1. (a) [1] Version 1: Fill in the blank: The implication q → p is the of the implication p → q . Solution: converse Version 2: Fill in the blank: The implication ¬ p → ¬ q is the of the implication p → q . Solution: inverse (b) [2] Version 1: Write down one of the associative laws of logic in symbolic form. Solution: p ∧ ( q ∧ r ) ⇔ ( p ∧ q ) ∧ r or p ∨ ( q ∨ r ) ⇔ ( p ∨ q ) ∨ r Version 2: Write down one of the distributive laws of logic in symbolic form. Solution: p ∨ ( q ∧ r ) ⇔ ( p ∨ q ) ∧ ( p ∨ r ) or p ∧ ( q ∨ r ) ⇔ ( p ∧ q ) ∨ ( p ∧ r ) (c) [6] Use the laws of logic and rules of inference to prove the validity of the following argument. Justify each step with the rules of inference and/or the laws of logic. Do NOT use a truth table. Version 1: p → ( q ∨ r ) q → s p ¬ s r Solution: Steps Reasons 1. p → ( q ∨ r ) Premise 2. q → s Premise 3. p Premise 4. ¬ s Premise 5. q ∨ r By 1, 3, and Modus Ponens 6. ¬ q By 2, 4, and Modus Tollens 7. r By 5, 6, and Rule of Disjunctive Syllogism