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Information about Math 189 Exam 1 Solutions

Jason Smith Exam 1 Solutions for Math 189, Boise State University, Spring 2014

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1.5 E. If p → q is false, then the truth value of (¬p ∨ ¬q) → (p ↔ q) is also false. True. Realize that the only case where p → q is false is when p is true and q is false. So ¬p is false and ¬q is true. So (¬p ∨ ¬q) is true since ¬q is true. Also, (p ↔ q) means (p → q) ∧ (q → p). However, since p → q is false, then (p ↔ q) is false. Therefore, since the ﬁrst part of the implication(¬p∨¬q) is true and the second (p ↔ q) is false, we have that (¬p ∨ ¬q) → (p ↔ q) is false when p → q is false. 2 2.1 Prove that if a and b are odd integers, then a ∗ b is an odd integer. Proof. Assume that a, b are odd integers. Then a = 2k + 1 for some integer k. Also, b = 2m + 1 for some integer m. We have that a ∗ b = (2k + 1)(2m + 1). = 4km + 2k + 2m + 1. = 2(2km + m + k) + 1 . Since k, m are integers, so is 2km + k + m. So a ∗ b is odd. 3 3.1 For any mathematical statement, say C, fn (C) denotes the mathematical statement: ¬¬ . . . ¬C, where there are n¬ symbols in front of C. Prove that if A is a True mathematical statement and B is a False mathematical statement, then ¬(f3 (A) ∨ f2 (B)) is a True mathematical statement. 2

Proof. Assume that A is a True mathematical statement and B is a False mathematical statement. We have that f3 (A) = ¬¬¬A = ¬(¬¬)A = ¬A since every pair of ¬ symbols cancels out by the Double Negative Law. We also have f2 (B) = ¬¬B = (¬¬)B = B. So we have ¬(¬A ∨ B), which is (¬¬A ∧ ¬B). So we have A ∧ ¬B. Since A is true and B is false (that is, ¬B is true), we have two true statements. The joining of two true statements with ”and” is true, so ¬(f3 (A) ∨ f2 (B)) is a True mathematical statement 4 4.1 4.1.1 p T T T T F F F F Find the Disjunctive Normal Form of ((p → q) ∧ (q → r)) → (p → r) Method: Truth Table q T T F F T T F F r T F T F T F T F ¬p F F F F T T T T ¬q F F T T F F T T ¬r F T F T F T F T p→q T T F F T T T T q→r T F T T T F T T p→r T F T F T T T T ((p → q) ∧ (q → r)) T F F F T F T T ((p → q) ∧ (q → r)) → (p → r) T T T T T T T T So we have (p ∧ q ∧ r) ∨ (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ r) ∨ (p ∧ ¬q ∧ ¬r) ∨ (¬p ∧ q ∧ r) ∨ (¬p ∧ q¬r) ∨ (¬p ∧ ¬q ∧ r) ∨ (¬p ∧ ¬q ∧ ¬r). Note that this the original statement is true no matter what the truth values of p, q, r. So Disjunctive Normal Form addresses each of the possible cases represented in the truth table. 3

5 5.1 Let a, b, c be real numbers. Prove that if a + b ≥ c, then a ≥ b ≥ c2. c 2 or Note: This can be proven directly but then you must address each of the following cases: c c (i). a = 2 , b = 2 . c c (ii) a = 2 , b > 2 . c c (iii) a > 2 , b = 2 (similar to ii) c c (iiii) a > 2 , b > 2 . This is why contradiction proofs are easier to use in situations where you have OR statements to deal with. In this situation, a contradiction gives us an AND statement, which is a single case. When writing proofs it is often helpful to ask yourself ”When can this be false?” Then show that that case cannot happen. So, c c Proof. Suppose for contradiction that a + b = c and a < 2 and b < 2 . c c Then a + b < 2 + 2 . So a + b < c, which contradicts our original assumption that a + b = c. 4

Jason Smith Exam 1 Solutions for Math 189, Boise State University, Spring 2014

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