Mat 251 Module One Exam Summer 2017 Due
Mat 251 Module One Exam Summer 2017 Due
MAT 251 Module One Exam Summer 2017 Due Sunday, June 18 at 11:59 pm. This is a firm deadline, so please plan accordingly to finish on time and make sure (in advance) that your access to a scanner is reliable. Answer all questions on separate paper and show all work wherever possible. PLEASE USE BLUE OR BLACK INK AND PLEASE MAKE SURE YOUR PRINTING IS DARK.
Paper For Above instruction
This exam comprises a series of logical and mathematical problems encompassing propositional logic, quantifiers, proof strategies, and number theory. The questions aim to evaluate your proficiency in translating everyday statements into symbolic logic, constructing and analyzing truth tables, and applying proof techniques such as direct proof, proof by contrapositive, contradiction, and induction. Additionally, you are expected to demonstrate the ability to find counterexamples and accurately negate statements with quantifiers, as well as perform algebraic and numerical reasoning related to integers, divisibility, and sums.
Answer the following questions comprehensively and clearly, demonstrating all necessary work, reasoning, and justifications.
1. Logical Statements and Contrapositives
a. Using the symbols p: You play the game and q: You win, express the sentence “If you don’t play the game, you don’t win” using symbolic logic.
b. Find the contrapositive of the sentence in part a). Express your answer both in words and in symbolic form.
2. Counterexamples in Number Theory
a. Find a counterexample to disprove the following statement: For all n ≥ 1, m + m− n is divisible by 4.
b. Find a counterexample to disprove the following statement: For all n ∈ ℕ, 2,9n− not divisible by 3.
3. Negating Quantified Statements
a. Find the negation of the statement: ∀x ∈ ℝ, 0 ≤ x + 2.
b. Find the negation of the statement: ∃m ∈ ℝ such that 10m + 7 = 10.
4. Logical Equivalences Using Truth Tables
a. Use truth tables to prove that p → q is logically equivalent to ¬p ∨ q.
b. Use truth tables to prove that (p ∨ q) is logically equivalent to (p ∧ q).
5. Constructing a Truth Table
Complete a truth table for the logical expression: (p ∧ q) ∨ (¬p ∧ r) ↔ (p ∧ q) ∨ (¬p ∧ r).
6. Number Theory and Divisibility
Use a direct proof to prove that the sum of any five consecutive odd integers is divisible by 5.
7. Proof by Contrapositive
Use a proof by contrapositive to prove: Let m and n be integers. Prove that if mn is even, then either m is even or n is even.
8. Proof by Contradiction
Use a proof by contradiction to prove: The number 7 is irrational.
9. Mathematical Induction
Use mathematical induction to prove: For all n ≥ 1, n + (n + 1) + ... + (2n) = n(n + 1)/2.
10. Proof by Cases or Mathematical Induction
Use a proof by cases or induction to prove: For all n ≥ 1, 5n − 5 + n is divisible by 5.
References
- Enderton, H. B. (2001). A Mathematical Introduction to Logic. Academic Press.
- Ross, K. A. (2014). Introduction to Probability and Statistics for Engineers and Scientists. Elsevier.
- Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
- Rosen, K. H. (2012). Discrete Mathematics and Its Applications. McGraw-Hill Education.
- Velleman, D. J. (2006). How to Prove It: A Structured Approach. Cambridge University Press.
- Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
- Epp, S. S. (2011). Discrete Mathematics with Applications. Brooks Cole.
- Stair, R., & Smith, R. (2016). Mathematical Proofs: An Introduction. Springer.
- Krantz, S. G. (2013). The History of Mathematics: A Brief Course. Oxford University Press.
- Hammack, R. (2018). Book of Proof. Cambridge University Press.