A Practical Introduction to Data Structures and Algorithm Analysis

28 Chap. 2 Mathematical PreliminariesExample 2.1 For the integers, = is an equivalence relation that partitionseach element into a distinct subset. In other words, for any integer a, threethings are true.1. a = a,2. if a = b then b = a, and3. if a = b and b = c, then a = c.Of course, for distinct integers a, b, and c there are never cases wherea = b, b = a, or b = c. So the claims that = is symmetric and transitive arevacuously true (there are never examples in the relation where these eventsoccur). But because the requirements for symmetry and transitivity are notviolated, the relation is symmetric and transitive.Example 2.2 If we clarify the definition of sibling to mean that a personis a sibling of him- or herself, then the sibling relation is an equivalencerelation that partitions the set of people.Example 2.3 We can use the modulus function (defined in the next section)to define an equivalence relation. For the set of integers, use the modulusfunction to define a binary relation such that two numbers x and y arein the relation if and only if x mod m = y mod m. Thus, for m = 4,〈1, 5〉 is in the relation because 1 mod 4 = 5 mod 4. We see that modulusused in this way defines an equivalence relation on the integers, and this relationcan be used to partition the integers into m equivalence classes. Thisrelation is an equivalence relation because1. x mod m = x mod m for all x;2. if x mod m = y mod m, then y mod m = x mod m; and3. if x mod m = y mod m and y mod m = z mod m, then x modm = z mod m.A binary relation is called a partial order if it is antisymmetric and transitive. 2The set on which the partial order is defined is called a partially ordered set or aposet. Elements x and y of a set are comparable under a given relation if either2 Not all authors use this definition for partial order. I have seen at least three significantly differentdefinitions in the literature. I have selected the one that lets < and ≤ both define partial orders on theintegers, because this seems the most natural to me.