Mathematical Foundations of Computer Science

2016

Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.

The Student Mathematical Library, 2002

The book is based on lectures given by the authors to undergraduate students at Moscow State University. It explains basic notions of "naive" set theory (cardinalities, ordered sets, transfinite induction, ordinals). The book can be read by undergraduate and graduate students and all those interested in basic notions of set theory. The book contains more than 100 problems of various degrees of difficulty.

Q1. Explain what it means for one set to be a subset of another set. How do you prove that one set is a subset of another set? A1: The set A is a subset of B if and only if every element of A is also an element of B. We use the notation A ⊆ B to indicate that A is a subset of the set B. We see that A ⊆ B if and only if the quantification ∀x(x ∈ A → x ∈ B) is true. Note that to show that A is not a subset of B we need only find one element x ∈ A with x /∈ B. Such an x is a counterexample to the claim that x ∈ A implies x ∈ B. We have these useful rules for determining whether one set is a subset of another: Showing that A is a Subset of B. To show that A ⊆ B, show that if x belongs to A then x also belongs to B. Showing that A is Not a Subset of B. To show that A _⊆ B, find a single x ∈ A such that x _∈B. Q2. What is the empty set? Show that the empty set is a subset of every set. A2: The empty set is the set with no elements. It satisfies the definition of being a subset of every set vacuously. Q3. a) Define |S|, the cardinality of the set S. b) Give a formula for |A ∪ B|, where A and B are sets. A3: a) Showing Two Sets are Equal. To show that two sets A and B are equal, show that A ⊆ B and B ⊆A. b) Sets may have other sets as members. For instance, we have the sets A = {∅, {a}, {b}, {a, b}} and B = {x | x is a subset of the set {a, b}}. Note that these two sets are equal, that is, A = B. Also note that {a} ∈ A, but a /∈ A. Q4. a) Define the power set of a set S. b) When is the empty set in the power set of a set S? c) How many elements does the power set of a set S with n elements have? A4: a) Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S). b) Always c) 2 n Q5. a) Define the union, intersection, difference, and symmetric difference of two sets. b) What are the union, intersection, difference, and symmetric difference of the set of positive integers and the set of odd integers? A5:a) Union: Let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set that contains those elements that are either in A or in B, or in both. An element x belongs to the union of the sets A and B if and only if x belongs to A or x belongs to B. This tells us that (A ∪ B = {x | x ∈ A ∨ x ∈ B}). E.g. The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, {1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}. Intersection: Let A and B be sets. The intersection of the sets A and B, denoted by A ∩ B, is the set containing those elements in both A and B. An element x belongs to the intersection of the sets A and B if and only if x belongs to A and x belongs to B. This tells us that (A ∩ B = {x | x ∈ A ∧ x ∈ B}). E.g. The intersection of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 3}; that is, {1, 3, 5} ∩ {1, 2, 3} = {1, 3}. b) Union: integers that are odd or positive, intersection: odd positive integers, difference: even positive integers, symmetric difference: even positive integers together with odd negative integers

there is no abstract here

Notation. We will use the sets N = {0, 1, 2, , 3,. . .} Z = {0, 1, −1, 2, −2,. . .}. 1.1. Induction First principle. Let P(n) be some statement that makes sense for all n n 0. (Typically, n 0 = 0, 1 or 2.) Suppose that (1) P(n 0) is true, and (2) for all n n 0 , P(n) is true ⇒ P(n + 1) is true. Then P(n) is true for all n. Example. P(n) is the assertion that 1 + 3 + 5 + · · · + (2n − 1) = n 2. Take n 0 = 1. (1) P(1) is true because both sides equal 1. (2) Now suppose that P(n) is true, and add 2n + 1 to both sides above to give 1 + 3 + 5 + · · · + (2n − 1) + (2n + 1) = n 2 + (2n + 1). The right-hand side simplifies to (n + 1) 2 , so this is assertion P(n + 1). Therefore P(n) must be true for all n 1. Note. Curly P emphasizes that P is a statement, not an arithmetical function. Second principle. Same start as above. Suppose that (1) P(n 0) is true, and (2') for all n n 0 , P(k) is true for all n 0 k < n ⇒ P(n) is true. Then P(n) is true for all n. Example. Take n 0 = 2. P(n) is the assertion " n can be written as a product of (one or more) prime numbers ". (1) 2 is a prime number, so obviously P(2) is true. (2') (i) If n is prime, then P(n) is already true. (ii) If not, then n has a divisor other than 1 and n, so we can write n = ab with 1 < a < n and 1 < b < n. If P(k) is true for all k < n then P(a) and P(b) are both true, which means that a is a prime or a product of primes, and b similarly. The same must be true of ab, and P(n) is true.

A theory of recursive definitions has been mechanized in Isabelle''s Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs, and other computational reasoning. Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski theorem over a suitable set.Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion.Recursive data structures are expressed by applying the Knaster-Tarski theorem to a set, such asV , that is closed under Cartesian product and disjoint sum. Worked examples include the transitive closure of a relation, lists, variable-branching trees, and mutually recursive trees and forests. The Schröder-Bernstein theorem and the soundness of propositional logic are proved in Isabelle sessions.

Springer eBooks, 1993

Jacobs University Bremen, 2004