Proof by contradiction set theory pdf

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. One way to prove a b is to assume that a is true and b is false. For a set of premises and a proposition, it is true that. In earlier versions of our logic style manual, we tended to. Weve got our proposition, which means our supposition is the opposite. The empty set is a subset of a, hence it is an element of the power set of a. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. Noncontradiction is also a theorem in propositional logic. 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. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. The objects in a set s are called the members of s. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory.

Proof, sets, and logic department of mathematics boise state. In standard introductory classes in algebra, trigonometry, and calculus there is currently very little emphasis on the discipline of proof. Contradiction proofs this proof method is based on the law of the excluded middle. Perhaps the most obvious thing to say is that there is an epistemological relationship between the two, such that. Occasionally there are situations where this method is not applicable. In mathematics, a contradiction is the assertion of a statement and its negation, or equivalently, a. They are not guaranteed to be comprehensive of the material covered in the course. By a wellfounded set, we mean a set ssuch that every sequence s s 0 3s 1 3s 2 3is nite. Set theory proof a is a subset of b if and only if a intersect b a broll islandesque by kevin macleod is licens. Its a principle that is reminiscent of the philosophy of a certain fictional detective.

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. Chapter 17 proof by contradiction university of illinois. Let set sbe the set of numbers which cannot be written as a product of primes. Hilbert viewed the axiomatic method as the crucial tool for mathematics and rational discourse in general. In that proof we needed to show that a statement p. That is, it is possible to determine if an object is to be included in the set or not. Set theory \a set is a many that allows itself to be thought of as a one. This alone assures the subject of a place prominent in human culture. I this video i prove the statement the sum of two consecutive numbers is odd using direct proof, proof by contradiction, proof by induction.

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. To prove that p is true, assume that p is not true. See proof 2 is section 5 for a proof by contradiction of n2 is even n is even. Introduction to logic and set theory202014 general course notes december 2, 20. 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. 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. March 30, 2009 basic set theory we will use the standard notation for containments. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen. The sum of two positive numbers is always positive. Russells paradox is the most famous of the logical or set theoretical paradoxes.

If p leads to a contradiction, then p cant be true, and therefore p must be true. Such a set appears to be a member of itself if and only if it is not a member of itself. Russells paradox stanford encyclopedia of philosophy. Proofs and mathematical reasoning university of birmingham. 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. 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. 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. Proof by contradiction this is an example of proof by contradiction. That is, a proof is a logical argument, not an empir. This states that an assertion or mathematical statement cannot be both true and false. The reason is that the proof setup involves assuming. X for some ordinal then the set z of elements of x. This text is for a course that is a students formal introduction to tools and methods of proof. 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.

In general, the context of the problem determines u. Set theory is indivisible from logic where computer science has its roots. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. Assume snot empty, so it has a smallest element nby wop. Proof by contradiction a proof by contradiction is a proof that works as follows. In this case, we implicitly assume all of number and set theory to tackle the problem. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. The idea of proof by contradiction is quite ancient, and goes back at least as far as the. Proof is, however, the central tool of mathematics. Review of set theory michael williams last updated. Please bear in mind that these are only examples of the vast power.

In classical logic, particularly in propositional and firstorder logic, a proposition is a contradiction if and only if. Basic set theory members of the collection comprising the set are also referred to as elements of the set. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Proof by contradiction is another important proof technique. 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. Proof by contradiction is typically used to prove claims that a certain type.

Proof by contradiction is based on the law of noncontradiction as first formalized as a metaphysical principle by aristotle. Anderson and greg welty what is the relationship between the laws of logic and the existence of god. If we want to prove a statement s, we assume that s wasnt true. The book starts with the basics of set theory, logic and truth tables, and counting. The same paradox had been discovered in 1899 by ernst zermelo but he did not publish the idea, which. The sum of two positive numbers is not always positive.

Consider the complementary set scwhose elements are the natural numbers that are not elements of s. The theory of numbers provides an excellent source for such examples. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Also, remember that when you choose to use proof by contradiction, you must show how. It goes without saying that a basic understanding of set theory is. The universal set is the set that contains everything concerned, usually denoted as u. New to proving mathematical statements and theorem. Many of the statements we prove have the form p q which, when negated, has the form p. We call proofs arguments and you should be convincing the. The proof by contradiction is grounded in the fact that any proposition must be either true or. The idea is to assume the hypothesis, then assume the.

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. Note the difference between the statements such a set does not exist and it is an empty set. In other words, either sc has a smallest element or else it is the empty set. 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. 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. For any set a, the empty set is an element of the power set of a. Set theory for computer science university of cambridge.

This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. Set theory proof a is a subset of b if and only if a. Is my proof by contradiction about the empty set correct. Chapter 6 proof by contradiction mcgill university. Proof by contradiction example 2 show that 2 is irrational. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. The cartesian product of two sets is a set, and the elements of that set are ordered pairs. Further, we may restrict a and b to have no common factor. But even more, set theory is the milieu in which mathematics takes place today. Based on the assumption that p is not true, conclude something impossible. Proof methods mathematical and statistical sciences. Not sure if this set theory proof attempt involving contradiction is valid. Using this assumption we try to deduce a false result, such as 0 1.

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. Then it can be expressed as a b, for some positive integers a and b with b 0. Mat231 transition to higher math proofs involving sets fall 2014 8 11. There exist two positive numbers a and b that sum to a negative number. Set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. A set is a collection of objects, which are called elements or members of the set. If all our steps were correct and the result is false, our initial assumption must have been wrong. Proof by contradiction albert r meyer contradiction. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. 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. Assume 0 proof of this corollary illustrates an important technique called proof by contradiction. Buss departments of mathematics and computer science, university of california, san diego. Assume 0 set a, the empty set is an element of the power set of a.

Then, the book moves on to standard proof techniques. If a set contains an ordinal, then it contains a minimal ordinal. 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. In other words, you assume that the hypothesis is true but the conclusion is false. Basic proof techniques washington university in st.

If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. This is also a set of natural numbers, to which we will apply the wellordered axiom. The proof by contradiction is grounded in the fact that any proposition must. The proof began with the assumption that p was false, that is that. Four basic proof techniques used in mathematics youtube. Each illustrates an aspect of set theory that youve learned so far. 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.

Call sa pure set i every such sequence ends with a set s. Problems on discrete mathematics1 ltex at january 11, 2007. Alternatively, you can do a proof by contradiction. Please refer to section 10, where the operations on sets are explained in detail.