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“Lower Bounds on Information Transfer in Distributed Computations,”

Mathematical Preliminaries In this section we introduce basic concepts used throughout the book. Since it is presumed that the reader has already met most of this material, this presentation is abbreviated. 1.2.1 Sets A set A is a non-repeating and unordered collection of elements. For example, A50s = {Cobol, Fortran, Lisp} is a set of elements that could be interpreted as the names of languages designed in the 1950s. Because the elements in a set are unordered, {Cobol, Fortran, Lisp} and {Lisp, Cobol, Fortran} denote the same set. It is very convenient to recognize the empty set ∅, a set that does not have any elements. The set B = {0, 1} containing 0 and 1 is used throughout this book. The notation a ∈ A means that element a is contained in set A. For example, Cobol ∈ A50s means that Cobol is a language invented in the 1950s. A set can be finite or infinite. The cardinality of a finite set A, denoted |A|, is the number of elements in A. We say that a set A is a subset of a set B, denoted A ⊆ B, if every element of A is an element of B. If A ⊆ B but B contains elements not in A, we say that A is a proper subset and write A ⊂ B. The union of two sets A and B, denoted A ∪ B, is the set containing elements that are in A, B or both. For example, if A0 = {1, 2, 3} and B0 = {4, 3, 5}, then A0 ∪ B0 = {5, 4, 3, 1, 2}. The intersection of sets A and B, denoted A∩B, is the set containing elements that are in both A and B. Hence, A0 ∩ B0 = {3}. If A and B have no elements in common, denoted A ∩ B = ∅, they are said to be disjoint sets. The difference between sets A and B, denoted A − B, is the set containing the elements that are in A but not in B. Thus, A0 − B0 = {1, 2}. (See Fig. 1.1.) A B A ∩ B A − B Figure 1.1 A Venn diagram showing the intersection and difference of sets A and B. Their union is the set of elements in both A and B. 8 Chapter 1 The Role of Theory in Computer Science Models of Computation The following simple properties hold for arbitrary sets A and B and the operations of set union, intersection, and difference: A ∪ B = B ∪ A A ∩ B = B ∩ A A ∪ ∅ = A A ∩ ∅ = ∅ A − ∅ = A The power set of a set A, denoted 2A, is the set of all subsets of A including the empty set. For example, 2{2,5,9} = {∅, {2}, {5}, {9}, {2, 5}, {2, 9}, {5, 9}, {2, 5, 9}}. We use 2A to denote the power set A as a reminder that it has 2|A| elements. To see this, observe that for each subset B of the set A there is a binary n-tuple (e1, e2, … , e|A|) where ei is 1 if the ith element of A is in B and 0 otherwise. Since there are 2|A| ways to assign 0’s and 1’s to (e1, e2, … , e|A|), 2A has 2|A| elements.