A Practical Introduction to Data Structures and Algorithm Analysis

50 Chap. 2 Mathematical Preliminariesalso discusses the three proof techniques presented in Section 2.6, and the roles ofinvestigation and argument in problem solving.For more about estimating techniques, see two Programming Pearls by JohnLouis Bentley entitled The Back of the Envelope and The Envelope is Back [Ben84,Ben00, Ben86, Ben88]. Genius: The Life and Science of Richard Feynman byJames Gleick [Gle92] gives insight into how important back of the envelope calculationwas to the developers of the atomic bomb, and to modern theoretical physicsin general.2.9 Exercises2.1 For each relation below, explain why the relation does or does not satisfyeach of the properties reflexive, symmetric, antisymmetric, and transitive.(a) “isBrotherOf” on the set of people.(b) “isFatherOf” on the set of people.(c) The relation R = {〈x, y〉 | x 2 + y 2 = 1} for real numbers x and y.(d) The relation R = {〈x, y〉 | x 2 = y 2 } for real numbers x and y.(e) The relation R = {〈x, y〉 | x mod y = 0} for x, y ∈ {1, 2, 3, 4}.(f) The empty relation ∅ (i.e., the relation with no ordered pairs for whichit is true) on the set of integers.(g) The empty relation ∅ (i.e., the relation with no ordered pairs for whichit is true) on the empty set.2.2 For each of the following relations, either prove that it is an equivalencerelation or prove that it is not an equivalence relation.(a) For integers a and b, a ≡ b if and only if a + b is even.(b) For integers a and b, a ≡ b if and only if a + b is odd.(c) For nonzero rational numbers a and b, a ≡ b if and only if a × b > 0.(d) For nonzero rational numbers a and b, a ≡ b if and only if a/b is aninteger.(e) For rational numbers a and b, a ≡ b if and only if a − b is an integer.(f) For rational numbers a and b, a ≡ b if and only if |a − b| ≤ 2.2.3 State whether each of the following relations is a partial ordering, and explainwhy or why not.(a) “isFatherOf” on the set of people.(b) “isAncestorOf” on the set of people.(c) “isOlderThan” on the set of people.(d) “isSisterOf” on the set of people.(e) {〈a, b〉, 〈a, a〉, 〈b, a〉} on the set {a, b}.