The concluding five chapters are on the beautiful subject of combinatory logic, which is not only intriguing in its own right, but has important applications to computer science. Argonne National Laboratory is especially involved in these applications, and I am proud to say that its members have found use for some of my results in combinatory logic. This book does not cover such important subjects as set theory, model theory, proof theory, and modern developments in recursion theory, but the reader, after studying this volume, will be amply prepared for the study of these more advanced topics.
Request Inspection Copy. Author : Raymond M. This introduction to discrete mathematics is aimed at freshmen and sophomores in mathematics and computer science. It begins with a survey of number systems and elementary set theory before moving on to treat data structures, counting, probability, relations and functions, graph theory, matrices, number theory and cryptography. The end of each section contains problem sets with selected solutions, and good examples occur throughout the text.
Written in a clear, precise and user-friendly style, Logic as a Tool: A Guide to Formal Logical Reasoning is intended for undergraduates in both mathematics and computer science, and will guide them to learn, understand and master the use of classical logic as a tool for doing correct reasoning. It offers a systematic and precise exposition of classical logic with many examples and exercises, and only the necessary minimum of theory.
The book explains the grammar, semantics and use of classical logical languages and teaches the reader how grasp the meaning and translate them to and from natural language. It illustrates with extensive examples the use of the most popular deductive systems -- axiomatic systems, semantic tableaux, natural deduction, and resolution -- for formalising and automating logical reasoning both on propositional and on first-order level, and provides the reader with technical skills needed for practical derivations in them.
Systematic guidelines are offered on how to perform logically correct and well-structured reasoning using these deductive systems and the reasoning techniques that they employ. The year's finest mathematical writing from around the world This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics makes available to a wide audience many articles not easily found anywhere else—and you don't need to be a mathematician to enjoy them.
For any component relation C x, y on a set A, the following three conditions are all equivalent: 1 The generalized induction principle holds. Prove the above. Compactness We now turn to another principle that is quite useful in mathematical logic and set theory.
Consider a universe V in which there are denumerably many people. The people have formed various clubs. A club C is called maximal if it is not a proper subset of any other club.
Thus if C is a maximal club, if we add one or more people to the set C the resulting set is not a club. Would that change the answer to a? Again we consider the case that V contains denumerably many inhabitants, and we assume that we are given the additional information that a set S of people is a club if and only if every finite subset of S is a club.
Then it does follow that if there is at least one club, there is a maximal club; better yet, every club is a subset of a maximal club. The problem is to prove this. Here are the key steps: 1 Show that under the given conditions, every subset of a club is a club. Now, given any club C that exists, define the following infinite sequence C0, C1, C2,.
Then consider x1. It is obvious that each Cn is a club. Consider now an arbitrary denumerable set A and a property P of subsets of A. The property P is called compact if it is the case that for every subset S of A, P holds for S if and only if P holds for all finite subsets of S. In the last problem, we were given a denumerable universe V and a collection of subsets called clubs, and we were given that a subset S of V is a club if and only if all finite subsets of S are clubs, in other words, we were given that the property of being a club is compact, from which we could deduce that any club is a subset of a maximal club.
Now, there is nothing special about clubs that makes our argument go through. The same reasoning yields the following: Denumerable Compactness Theorem. For any compact property P of subsets of a denumerable set A, any subset S of A having property P is a subset of a maximal set that has property P. Note: The above result actually holds even if A is a non-denumerable set whether finite or infinite , but the more advanced result for infinite non-denumerable sets is not needed for anything in this book.
The reader should also be able to see that the proof just given can be modified slightly to work when V is finite. Discussion In mathematical logic, we deal with a denumerable set of symbolic sentences, some subsets of which are deemed inconsistent.
Now, since the mathematical logic we will be studying in this book considers only proofs employing a finite number of sentences, any proof that a given set of sentences is inconsistent uses only finitely many members of the set of those sentences.
Thus a denumerable set S is defined to be consistent if and only if all finite subsets of S are consistent.
In other words, consistency is a compact property, a fact that will prove very important later on! Solutions to the Problems of Chapter 4 1. If he were of type T, then he really would have said that sometime before, as he said, and when he said that before, he was still of type T, hence he had said it sometime before that, hence sometime before even that, and so forth. Thus unless he lived infinitely far back into the past, he cannot be of type T. Thus he is of type F.
We consider a property P such that for every number n, if P holds for all numbers less than n, then P also holds for n.
We are to show that P holds for all numbers. Now, there are no natural numbers less than 0, so that the set of natural numbers less than zero is empty. Thus, by our assumption of the induction premise of complete mathematical induction i. We consider a property Q stronger than P in the sense that any number having property Q also has property P , and we use mathematical induction on the property Q, i.
In fact we define Q n to mean that P holds for n and all numbers less than n. Of course Q n implies P n. Now suppose n is a number for which Q holds. We are then to infer the principle of mathematical induction.
We are to show that P holds for every n. To do this, it suffices to show that for any number n, if P holds for all numbers less than n. Well, suppose P holds for all numbers less than n. Since P holds for all numbers less than n, then P holds for m.
This proves that if P holds for all numbers less than n, then P holds for n. Then by the assumed principle of complete mathematical induction, P holds for all numbers. We are to then prove the least number principle.
Define P n to mean that any set A that contains a number x no greater than n must have a least element. Note: To say that A contains a number x no greater than n just means that A contains some number x that is less than or equal to n. First we show by mathematical induction that every number n has property P. Suppose A is a set that contains a number x that is no greater than zero.
Then x must be zero; hence 0 is an element of A, and thus the least element of A. This proves that P hold for 0. Next, suppose that P holds for n. We must show that A contains a least element. On the other hand, if it does, then it contains some number not greater than n, hence by our assumption of P n , A must contain a least element. Now let A be any non-empty set of numbers. Then A contains some number n. Thus, since P holds for n as we have seen , A must contain a least element.
The proves the least number principle. We wish to then prove the principle of mathematical induction. If P fails to hold for every n, then there is at least one number for which P fails. Hence, by the least number principle which we are assuming , there must be a least number n for which P fails. Since m is less than n and n is the least number for which P fails, then P cannot fail for m, and so P must hold for m. This is a contradiction, and so P cannot fail for any number. Thus P holds for every number.
We are to show that P holds for all x less than or equal to n. We will use mathematical induction on Q to show that Q holds for all x. Next, suppose that Q x holds. We are numbering the billiard balls B1,. The proof fails because the induction fails to go from 1 to 2. P 1 is true, but P 2 is false, since it is not true that any two billiard balls are of the same color indeed, if it were true, then all billiard balls would indeed be of the same color!
We are to prove by induction that every number y is special. Well, we are given that R x, 0 holds for every x; hence 0 is special. Now suppose that y is special. We show this by induction on x which is where an induction within an induction comes in! Thus R x, y holds for all x and y. We first show that every right normal number x is also left normal. Well, suppose x is right normal.
We show by induction on y that R x, y holds for all y. We are given outright that R x, 0 holds. Now suppose that y is such that R x, y holds. Since x is assumed right normal, then R y, x holds. And since R x, 0 holds, then by induction R x, y holds for all y, which means that x is left normal. Now we show by mathematical induction that every number y is right normal hence also left normal. By 1 , R x, 0 holds for all x, which tells us that 0 is right normal. Now suppose that y is right normal.
Thus, since every number x is both left and right normal, R x, y holds for every x and y. The answer is that the box must become empty sooner or later.
One proof of this, which we now give, is by mathematical induction. Another proof will be given later on. Call a positive integer n a losing number if it is the case that every ball game in which every ball originally in the box is numbered n or less i. We will now show by mathematical induction that every positive integer n is a losing number.
Now suppose that n is a losing number. One cannot keep throwing out balls numbered n or less forever, since n is assumed to be a losing number.
And from then on, by the induction assumption that n is a losing number, the process must eventually terminate. This completes the proof that every positive integer n is a losing number, which means that for any given box, no matter what daily choices are made, the process must eventually terminate.
As previously remarked, another proof will be given later, one which I believe is far neater and more elegant than the one just given. The curious thing about this ball game is that there is no finite limit to how long the player can live assuming that at least one initial ball is numbered 2 or higher , yet the player cannot live forever. It is interesting that many people give the wrong answer to this question.
They believe that there must have to be an infinite path, whereas in fact, there does not have to be one, as the following tree illustrates: The origin of this tree has denumerably many successors numbered 1, 2,. Every other point of the tree has only one successor.
The path through point number 1 stops at level 1. The path through point 2 goes down only two levels, and so forth. That is, for each n, the path through point n stops at level n. Thus, for each n, there is a path of length n, yet none of the paths are infinite. The situation can be likened to the natural numbers. Obviously for each natural number n, there is a number equal to or greater than n, yet no natural number is itself infinite. Now suppose n is such that there are only finitely many points x1,.
Let n1 be the number of successors of x1,. Thus, by mathematical induction, every level contains only finitely many points. It is obvious that statement 1 implies statement 2 , since statement 1 implies that every level n has at least one point.
If there were some n such that no path was of length n, then only finitely many levels would have any points at all since if there is no path of length n, there can be no points at level n or higher. And by a each of these levels would have only finitely many points.
Thus there would be only finitely many points on the tree, contrary to the assumed condition that there are infinitely many points on the tree.
Call a point rich if it has infinitely many descendants, and poor if it does not. We recall that if a point x has only finitely many successors, if each of those successors is poor, then x must be poor. Thus if x is rich and has only finitely many successors, then it cannot be that all its successors are poor. At least one of them must be rich. Thus in a finitely generated tree, each rich point must have a rich successor. For example, in the tree of the solution of Problem 10, the origin is rich, but all its successors are poor.
Since there are infinitely many points, and all points other than the origin are successors of the origin, the origin must be rich, hence must have at least one rich successor x1 as we have seen , which in turn has at least one rich successor x2, which in turn has at least one rich successor x3, and so forth. We thus generate an infinite path. Therefore the tree cannot be infinite. It must be finite.
To each ball game we associate a tree as follows: We take the origin to be a ball of any higher number than any of the balls initially in the box. We take the successors of the origin to be the balls that are initially in the box. For any ball x that is ever in the box, we take its successors to be the balls that replace it if any ever do. Since every ball that is ever replaced is replaced by only finitely many balls, the tree is finitely generated.
Since all replacements of a ball have lower numbers, each path must be finite. Then by the Fan Theorem, the tree is finite. Hence only finitely many balls ever find themselves in the box if only for a time.
And that would make the tree we constructed infinite, since it contains all balls that were ever in the box. Remember we assumed that once a ball was thrown out of the box, that particular ball is never put back into the box, so that each ball thrown out must be different from every other ball thrown out. If an inductive property P fails for x, then it must fail for some component x1 of x, hence it must also fail for some component x2 of x1, and so forth; which would mean that there is an infinite descending chain.
Thus if all descending chains are finite, an inductive property cannot fail for any element, and thus holds for all elements of A. We are considering a tree which is finitely generated and is such that all its paths are finite. We must show that the tree itself must be finite. For any point x of the tree, we define its components to be its successors. Let P x be the property that x is poor has only finitely many descendants. The descending chains of this component relation are the paths of the tree, and we are given that they are all finite.
Hence, by the Generalized Induction Theorem, P holds for all points of the tree. In particular, P holds for the origin, so that the origin is poor. Hence there are only finitely many points on the tree. It is obvious that if no component of x begins an infinite descending chain, neither does x since any chain starting with x has to first go through a component of x. Thus the property of not beginning any infinite chain is inductive. Thus if the generalized induction principle holds, then all elements have this property, which means that no element x begins an infinite descending chain, and so all descending chains are finite.
We will show that well-foundedness is equivalent to the condition that there are no infinite descending chains a In one direction, suppose a component relation C x, y on a set A is well-founded. If there were an infinite descending chain, then the subset of A composed of the elements of the chain would have no initial element, contrary to the given condition that the relation C is well-founded. Given any non-empty subset S of A, let x1 be any element of S. If x1 has no components in S, we are done; otherwise, x1 has a component x2 in S.
If x2 is an initial element of S, we are done; otherwise, x2 has a component x3 in S, and so forth. Since there are no infinite descending chains, we must sooner or later come to an initial element xn of S. Thus the component relation is well-founded. For example, it could be that all and only the finite sets are the clubs, and there is obviously no maximal finite set. If C is maximal, we are done. Otherwise C is a proper subset of a club C1. If C1 is maximal, we are done.
If not, then C1 is a proper subset of some club C2, and so forth. This sequence of clubs cannot go on forever, since V is finite.
Thus it must end in some maximal club Cn. We are now given that a set is a club if and only if all its finite subsets are clubs. Well, consider any subset S of C. Obviously, all subsets of S are also subsets of C. Now let F be any finite subset of S. Since F is clearly also a finite subset of C, it must be a club. Thus all finite subsets of S are clubs; hence S is a club.
Each member of F belongs to one of the clubs C0, C1, C2,. Of the finitely many clubs just chosen from the sequence, at least one of them includes all the others namely any one for which no other club chosen has a larger subscript.
Hence that club alone contains all members of F, making F a subset of a club Cn for some n. And since every subset of a club is a club as shown in 1 , F must be a club. To see this, let A be any subset of S. The starting point is Propositional Logic, which is the basis of First-Order Logic, as well as of higher order logics. Propositions can be combined to form more complex propositions by using the so-called logical connectives.
These two facts are summarized in the following table, which is called the truth table for negation. Each row in a truth table corresponds to a particular distribution of truth values for the variables that occur in the formula that the truth table is for. The first row of this truth table for negation i.
It has the following truth table, to reflect the four possible cases: 1 p and q are both true; 2 p is true and q is false; 3 p is false and q is true; 4 p and q are both false.
In Chapter 1 I gave an example to offer some justification for this seemingly strange use, and here is another: Consider the following proposition about the natural numbers. Formulas To approach our subject more rigorously, we must define the notion of a formula. The letters p, q, r, with or without subscripts, are called propositional variables.
By a formula is meant any expression constructed according to the following rules: 1 Each propositional variable is a formula. It is to be understood that no expression is a formula unless its being so is a consequence of rules 1 and 2. Show that the set of all formulas of Propositional Logic is a denumerable set. We can define matching parentheses in a formula in two ways which come down to being the same thing : 1 If we can see clearly how a formula is built up from its innermost propositional variables and connectives and parentheses, then, at each point that a pair of parentheses is added around a newly constructed component of the formula, those two parentheses consist in a pair of matching parentheses.
When two parentheses are matching, we also say they match each other. The second definition of matching parentheses above suggests how to pair off the matching parentheses in a complex formula one might be given. One first pairs all parenthesis pairs in which a right parenthesis follows a left parenthesis and there are no parentheses between them. Then one continues pairing the remaining unmatched parentheses in the resulting formula in the same way.
Incidentally, if, at the end of doing this, some parentheses remain unpaired, that means the original formula was badly formed, i. A way people often write complex formulas to help them to be understandable as well as to help checking that they are well-formed, is to use different kinds of parentheses within the formula, i.
These can make matching parentheses more apparent. Compound Truth Tables By the truth value of a proposition p is meant its truth or falsity, that is, the truth value of p is T if p is true, and is F if p is false. Therefore, given any combination of p and q, that is, given any proposition expressed in terms of p and q using the logical connectives, we can determine the truth value of the combination, given the truth values of p and q.
There are four distributions of truth values for p and q: p true, q true; p true, q false; p false, q true; p false, q false. We can determine the truth value of X in all these cases systematically by constructing the following table, which is an example of a compoundtruth table. We see that X is true in the first two cases and false in the last two. Such formulas are called tautologies. As we have seen, for a formula with n variables, there are 2n interpretations, each corresponding to a row in its truth table.
A formula that is neither a tautology nor a contradiction is called contingent. Author : Raymond M. Smullyan Publisher: Courier Corporation ISBN: Category: Mathematics Page: View: Read Now » Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts.
He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs.
Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and graduate courses, this book will also amuse and enlighten mathematically minded readers. Dover original publication. See every Dover book in print at www. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics makes available to a wide audience many articles not easily found anywhere else—and you don't need to be a mathematician to enjoy them.
These essays delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice—and taking readers behind the scenes of today's hottest mathematical debates.
In this volume, Moon Duchin explains how geometric-statistical methods can be used to combat gerrymandering, Jeremy Avigad illustrates the growing use of computation in making and verifying mathematical hypotheses, and Kokichi Sugihara describes how to construct geometrical objects with unusual visual properties. In other essays, Neil Sloane presents some recent additions to the vast database of integer sequences he has catalogued, and Alessandro Di Bucchianico and his colleagues highlight how mathematical methods have been successfully applied to big-data problems.
And there's much, much more. In addition to presenting the year's most memorable math writing, this must-have anthology includes an introduction by the editor and a bibliography of other notable writings on mathematics. This is a must-read for anyone interested in where math has taken us—and where it is headed. It serves not only as a tribute to one of the great thinkers in logic, but also as a celebration of self-reference in general, to be enjoyed by all lovers of this field.
Raymond Smullyan, mathematician, philosopher, musician and inventor of logic puzzles, made a lasting impact on the study of mathematical logic; accordingly, this book spans the many personalities through which Professor Smullyan operated, offering extensions and re-evaluations of his academic work on self-reference, applying self-referential logic to art and nature, and lastly, offering new puzzles designed to communicate otherwise esoteric concepts in mathematical logic, in the manner for which Professor Smullyan was so well known.
This book is suitable for students, scholars and logicians who are interested in learning more about Raymond Smullyan's work and life. Author : W. It begins with a survey of number systems and elementary set theory before moving on to treat data structures, counting, probability, relations and functions, graph theory, matrices, number theory and cryptography. The end of each section contains problem sets with selected solutions, and good examples occur throughout the text.
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