Exam 1 review 2021 PDF

Preview

Summary

Discrete exam review for the first exam...

Description

MAT 2358: Discrete Math I Exam 1 Review 1 What to expect: Exam 1 will occur during class on Tuesday, March 2, 2021. Exam 1 consists of five or six problems. You will have tables of logical equivalences and rules of inference to use during the exam.

2 How to study: Review the important concepts. Make sure you know how to properly apply the definitions of concepts. Go through practice problems – but! make sure you try to do them without looking at your notes or any references. Bring questions to office hours / Feel free to email Dr Lew with questions!

3 Concepts to Know • Propositional logic • Logical Equivalences • Predicates and Quantifiers (and Nested Quantifiers) • Even and odd integers • Using rules of inference to prove a conclusion • Proof methods: Direct Proofs, Proofs by Contraposition, Proofs by Contradiction, Proofs by Cases, Existence Proofs

4 Problems to study/review for the exam: • Any of the textbook homework problems (not just the suggested problems!) • Any problems from sections we’ve covered in the textbook • Any problems/examples we’ve discussed in class

1

MATH 2358 Exam date: Tuesday, March 2, 2021

5 Some Review Problems 1. Define (using truth tables) the disjunction, conjunction, exclusive or, conditional, and biconditional of the propositions p and q . 2. Give the disjunction, conjunction, exclusive or, conditional, and biconditional of the propositions “I’ll finish my discrete mathematics homework.” and “I’ll go to the mall this weekend.” 3. Define the inverse, converse, and contrapositive of a conditional statement. 4. Describe the two different ways to show that two compound propositions are logically equivalent. 5. Show in the two different ways that ¬p ∨ (r → ¬q) and ¬p ∨ ¬q ∨ ¬r are logically equivalent. 6. Give an example of the predicate P (x, y) such that ∃x∀yP (x, y) and ∀y∃xP (x, y) have different truth values. 7. Use the rules of inference to show that if ∀x(P (x) ∨ Q(x)) and ∀x((¬P (x) ∧ Q(x)) → R(x)) are true, then ∀x(¬R(x) → P (x)) is also true, where the domains of all quantifiers are the same. 8. Prove or disprove that the product of two irrational numbers is irrational. 9. Prove or disprove that the product of a nonzero rational number and an irrational number is irrational. 10. Prove or disprove that the difference of a nonzero rational number and an irrational number is rational. 11. Use a proof by contraposition to show that if x + y ≥ 2, where x and y are real numbers, then x ≥ 1 or y ≥ 1. 12. Prove that if n is an integers and 3n + 2 is even, then n is even using (a) a proof by contraposition and (b) a proof by contradiction. 13. Prove that if n is an odd integer, then there is a unique integer k such that n is the sum of k − 2 and k + 3.

2...