Proof by contradiction set theory pdf

Its a principle that is reminiscent of the philosophy of a certain fictional detective. In that proof we needed to show that a statement p. We call proofs arguments and you should be convincing the. Proof by contradiction a proof by contradiction is a proof that works as follows. The proof by contradiction is grounded in the fact that any proposition must. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. New to proving mathematical statements and theorem. One way to prove a b is to assume that a is true and b is false. This states that an assertion or mathematical statement cannot be both true and false. That is, a proof is a logical argument, not an empir. That is, it is possible to determine if an object is to be included in the set or not. This is also a set of natural numbers, to which we will apply the wellordered axiom. Alternatively, you can do a proof by contradiction.

Continued since each set is a subset of the other, we have established the equality of the two sets so a b c a b \a c. Russells paradox is the most famous of the logical or set theoretical paradoxes. The idea is to assume the hypothesis, then assume the. The sum of two positive numbers is not always positive. Proof by contradiction albert r meyer contradiction. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. In a talk to the swiss mathematical society in 1917, published the following year as axiomatisches denken 1918, he articulates his broad perspective on that method and presents it at work by considering, in detail, examples from various parts of.

Weve got our proposition, which means our supposition is the opposite. For a set of premises and a proposition, it is true that. The idea of proof by contradiction is quite ancient, and goes back at least as far as the. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Assume 0 set a, the empty set is an element of the power set of a. For any set a, the empty set is an element of the power set of a. Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. Assume 0 proof of this corollary illustrates an important technique called proof by contradiction.

A contradiction can be any statement that is wellknown to be false or a set of statements that are obviously inconsistent with one another, e. In classical logic, particularly in propositional and firstorder logic, a proposition is a contradiction if and only if. Review of set theory michael williams last updated. Proof by contradiction is typically used to prove claims that a certain type. Each illustrates an aspect of set theory that youve learned so far. If we want to prove a statement s, we assume that s wasnt true. March 30, 2009 basic set theory we will use the standard notation for containments. Chapter 17 proof by contradiction university of illinois. Sep 25, 2017 i this video i prove the statement the sum of two consecutive numbers is odd using direct proof, proof by contradiction, proof by induction and proof by contrapositive. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. The empty set is a subset of a, hence it is an element of the power set of a. Then it can be expressed as a b, for some positive integers a and b with b 0. Proof theory is concerned almost exclusively with the study of formal proofs.

In general, the context of the problem determines u. Contradiction proofs this proof method is based on the law of the excluded middle. Set theory \a set is a many that allows itself to be thought of as a one. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. Further, we may restrict a and b to have no common factor. This text is for a course that is a students formal introduction to tools and methods of proof. They are not guaranteed to be comprehensive of the material covered in the course. If all our steps were correct and the result is false, our initial assumption must have been wrong. See proof 2 is section 5 for a proof by contradiction of n2 is even n is even.

If a set contains an ordinal, then it contains a minimal ordinal. It is quite clear that most of these laws resemble or, in fact, are analogues of laws in basic algebra and the algebra of propositions. The sum of two positive numbers is always positive. We arrive at a contradiction when we are able to demonstrate that a statement is both simultaneously true and false, showing that our assumptions are inconsistent. X for some ordinal then the set z of elements of x which are ordinals is nonempty, and applying the axiom of foundation to this set we obtain an ordinal.

If p leads to a contradiction, then p cant be true, and therefore p must be true. Proof by contradiction is based on the law of noncontradiction as first formalized as a metaphysical principle by aristotle. Set theory proof a is a subset of b if and only if a intersect b a broll islandesque by kevin macleod is licens. The same paradox had been discovered in 1899 by ernst zermelo but he did not publish the idea, which. Basic proof techniques washington university in st. Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. I this video i prove the statement the sum of two consecutive numbers is odd using direct proof, proof by contradiction, proof by induction. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Consider the complementary set scwhose elements are the natural numbers that are not elements of s.

The use of this fact forms the basis of the technique of proof by contradiction, which mathematicians use extensively to establish the validity of a wide range of. Occasionally there are situations where this method is not applicable. Noncontradiction is also a theorem in propositional logic. It goes without saying that a basic understanding of set theory is. Four basic proof techniques used in mathematics youtube. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. Please bear in mind that these are only examples of the vast power.

Proof, sets, and logic department of mathematics boise state. Problems on discrete mathematics1 chungchih li2 kishan mehrotra3 syracuse university, new york latex at january 11, 2007 part i 1no part of this book can be reproduced without permission from the authors. The reason is that the proof setup involves assuming. The proof by contradiction is grounded in the fact that any proposition must be either true or. Using this assumption we try to deduce a false result, such as 0 1. By a wellfounded set, we mean a set ssuch that every sequence s s 0 3s 1 3s 2 3is nite. Also, remember that when you choose to use proof by contradiction, you must show how. The cartesian product of two sets is a set, and the elements of that set are ordered pairs. Common types of proofs disproof by counterexample statement must be of the form every x satisfies fx disprove it by finding some x that does not satisfy fx application of quantifier negation. Hilbert viewed the axiomatic method as the crucial tool for mathematics and rational discourse in general. Buss departments of mathematics and computer science, university of california, san diego. Basic set theory members of the collection comprising the set are also referred to as elements of the set. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect.

Dec, 2018 set theory proof a is a subset of b if and only if a intersect b a broll islandesque by kevin macleod is licensed under a creative commons attribution license. In this case, we implicitly assume all of number and set theory to tackle the problem. Assume snot empty, so it has a smallest element nby wop. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. This alone assures the subject of a place prominent in human culture. In earlier versions of our logic style manual, we tended to. X for some ordinal then the set z of elements of x. Proof is, however, the central tool of mathematics. Proofs and mathematical reasoning university of birmingham.

Set theory proof a is a subset of b if and only if a. In mathematics, a contradiction is the assertion of a statement and its negation, or equivalently, a. The objects in a set s are called the members of s. Proof by contradiction also known as indirect proof or the method of reductio ad absurdum is a common proof technique that is based on a very simple principle. Many of the statements we prove have the form p q which, when negated, has the form p. In other words, either sc has a smallest element or else it is the empty set. Perhaps the most obvious thing to say is that there is an epistemological relationship between the two, such that.

The theory of numbers provides an excellent source for such examples. The upshot of this discussion is that the paradoxes of set theory give no grounds for doubting the consistency of the theory of trans nite numbers or its embodiment in ordinary set theory. Introduction to logic and set theory202014 general course notes december 2, 20. The whole point of russells paradox is that the answer such a set does not exist means the definition of the notion of set within a given theory is unsatisfactory. Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen. Proof by contradiction is another important proof technique. In standard introductory classes in algebra, trigonometry, and calculus there is currently very little emphasis on the discipline of proof. Russells paradox stanford encyclopedia of philosophy. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory. Anderson and greg welty what is the relationship between the laws of logic and the existence of god. Please refer to section 10, where the operations on sets are explained in detail. Call sa pure set i every such sequence ends with a set s. Proof by contradiction example 2 show that 2 is irrational.

Set theory is indivisible from logic where computer science has its roots. Let set sbe the set of numbers which cannot be written as a product of primes. The book starts with the basics of set theory, logic and truth tables, and counting. Proof by contradiction this is an example of proof by contradiction. Mat231 transition to higher math proofs involving sets fall 2014 8 11. Problems on discrete mathematics1 ltex at january 11, 2007. Chapter 6 proof by contradiction mcgill university. But even more, set theory is the milieu in which mathematics takes place today. The universal set is the set that contains everything concerned, usually denoted as u. Unfortunately, not all proposed proofs of a statement in mathematics are actually correct, and so some e ort needs to be put into veri cation of such a proposed proof. Not sure if this set theory proof attempt involving contradiction is valid. To prove that p is true, assume that p is not true. Set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student.

Note the difference between the statements such a set does not exist and it is an empty set. Based on the assumption that p is not true, conclude something impossible. There exist two positive numbers a and b that sum to a negative number. Is my proof by contradiction about the empty set correct. Then, the book moves on to standard proof techniques. Such a set appears to be a member of itself if and only if it is not a member of itself. In other words, you assume that the hypothesis is true but the conclusion is false. Proof methods mathematical and statistical sciences. A set is a collection of objects, which are called elements or members of the set. The proof began with the assumption that p was false, that is that.