Download Practice Problems for Exam 1 - Discrete Structures | CMSC 250 and more Exams Discrete Structures and Graph Theory in PDF only on Docsity! 250, practice exam-I 1. Is √ d2e < ⌈ √ 2 ⌉ ? EXPLAIN. 2. What is ∑4 i=1 d(i + 10)/(i + 1)e? (note- each term has a ceiling.) 3. For each of the following arguments, either (1) write valid and show it is valid using only the rules of logic on the sheet (which are at the back of the exam), or (2) Write not valid and show why using rows of a truth table. (a) p1 : p1 → p2 p2 : p2 → p3 p3 : p3 → p4 p4 : p4 → p5 p5 : p5 → p6 p6 : p6 → p7 p7 : p7 → p1 conclusion : p2 → p1 (b) p1 : p1 ∨ p2 ∨ ¬p3 p2 : ¬p1 ∨ p3 p3 : p3 → p1 conclusion ¬p1 ∨ ¬p2 4. For this problem the symbols ≤, −, and || (the absolute value sign) have their usual meaning. for the sentence below, give (1) a non-empty domain where the sentence is true, and (2) a non-empty domain where the sentence is false. all domains given should be subsets of r (the reals). Justify your answers (one brief sentence each). (∃z)(∀x)(∀y)[|x− y| ≤ z]. 5. Prove the following. (a) for all naturals x, if x5 ≡ 0 (mod 3) then x ≡ 0 (mod 3). (b) show that 31/5 is irrational. you may not use unique factorization. (hint- use part a) 1 6. Mark as always true (a), sometimes true (s), or never true (n), for all real numbers x and all sequences ai: (a) x ≤ dxe (b) dxe − 1 ≤ x (c) dxe − 1 < x ≤ dxe (d) bxc < dxe (e) ∑n i=1 i = n + ∑n−1 i=1 i (f) ∑n i=1 ai = ∑n−1 i=0 ai (g) ∑n j=0 aj = ∑n+1 k=1 ak−1 7. The next term in the sequence 1, 3, 5, 7, ... is (a) certainly 9 (b) possibly 9 (c) certainly 11 (d) possibly 11 (e) both b and d (f) none of the above 8. In class, a property P ( ) was said to be hereditary (over the natural numbers) if for all n, P (n) → P (n + 1). For each of the following, say whether it is hereditary (DO NOT JUSTIFY): (a) P(n) means n is prime (b) P(n) means n is odd (c) P(n) means 1 + ... + n = n(n + 1)/2 (d) P(n) means n is either 0 or 1 or prime or composite (e) P(n) means n is irrational (f) P(n) means n2 + (n + 1)2 = (n + 2)2 9. The only way a conditional statement P → Q can be false is (a) P is false (b) Q is true (c) P is true and Q is false (d) none of the above 2
