Hilbert viewed the axiomatic method as the crucial tool for mathematics and rational discourse in general. Further, we may restrict a and b to have no common factor. The book starts with the basics of set theory, logic and truth tables, and counting. 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. In classical logic, particularly in propositional and firstorder logic, a proposition is a contradiction if and only if. 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. See proof 2 is section 5 for a proof by contradiction of n2 is even n is even. This text is for a course that is a students formal introduction to tools and methods of proof. Proof is, however, the central tool of mathematics. The reason is that the proof setup involves assuming. 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.
Perhaps the most obvious thing to say is that there is an epistemological relationship between the two, such that. 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. Proof by contradiction is another important proof technique. 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. Each illustrates an aspect of set theory that youve learned so far. Proofs and mathematical reasoning university of birmingham. The universal set is the set that contains everything concerned, usually denoted as u. For a set of premises and a proposition, it is true that. Weve got our proposition, which means our supposition is the opposite. 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. Problems on discrete mathematics1 ltex at january 11, 2007. 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. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. Mat231 transition to higher math proofs involving sets fall 2014 8 11. Please refer to section 10, where the operations on sets are explained in detail. In other words, you assume that the hypothesis is true but the conclusion is false.
To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. Assume 0 set a, the empty set is an element of the power set of a. Introduction to logic and set theory202014 general course notes december 2, 20. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever.
I this video i prove the statement the sum of two consecutive numbers is odd using direct proof, proof by contradiction, proof by induction. Basic proof techniques washington university in st. In earlier versions of our logic style manual, we tended to. The same paradox had been discovered in 1899 by ernst zermelo but he did not publish the idea, which. 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. 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. Let set sbe the set of numbers which cannot be written as a product of primes. Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. Consider the complementary set scwhose elements are the natural numbers that are not elements of s. Set theory \a set is a many that allows itself to be thought of as a one.
Set theory proof a is a subset of b if and only if a intersect b a broll islandesque by kevin macleod is licens. Proof by contradiction albert r meyer contradiction. 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. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Occasionally there are situations where this method is not applicable. If a set contains an ordinal, then it contains a minimal ordinal. In this case, we implicitly assume all of number and set theory to tackle the problem. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. This states that an assertion or mathematical statement cannot be both true and false. March 30, 2009 basic set theory we will use the standard notation for containments. Set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Not sure if this set theory proof attempt involving contradiction is valid. 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. 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.
They are not guaranteed to be comprehensive of the material covered in the course. Using this assumption we try to deduce a false result, such as 0 1. Noncontradiction is also a theorem in propositional logic. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. In general, the context of the problem determines u. If p leads to a contradiction, then p cant be true, and therefore p must be true. Contradiction proofs this proof method is based on the law of the excluded middle. By a wellfounded set, we mean a set ssuch that every sequence s s 0 3s 1 3s 2 3is nite. 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.
That is, a proof is a logical argument, not an empir. The proof by contradiction is grounded in the fact that any proposition must. Also, remember that when you choose to use proof by contradiction, you must show how. For any set a, the empty set is an element of the power set of a. In mathematics, a contradiction is the assertion of a statement and its negation, or equivalently, a. Proof, sets, and logic department of mathematics boise state. Russells paradox is the most famous of the logical or set theoretical paradoxes. This alone assures the subject of a place prominent in human culture. Russells paradox stanford encyclopedia of philosophy. The idea of proof by contradiction is quite ancient, and goes back at least as far as the. 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.
Review of set theory michael williams last updated. Four basic proof techniques used in mathematics youtube. The sum of two positive numbers is always positive. The sum of two positive numbers is not always positive. 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. 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. Proof by contradiction is based on the law of noncontradiction as first formalized as a metaphysical principle by aristotle. 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. In that proof we needed to show that a statement p. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Chapter 17 proof by contradiction university of illinois. The proof began with the assumption that p was false, that is that.
The idea is to assume the hypothesis, then assume the. Call sa pure set i every such sequence ends with a set s. 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. It goes without saying that a basic understanding of set theory is. 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. Buss departments of mathematics and computer science, university of california, san diego. Its a principle that is reminiscent of the philosophy of a certain fictional detective. In other words, either sc has a smallest element or else it is the empty set.
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. If we want to prove a statement s, we assume that s wasnt true. Then it can be expressed as a b, for some positive integers a and b with b 0.
The objects in a set s are called the members of s. Proof theory is concerned almost exclusively with the study of formal proofs. Chapter 6 proof by contradiction mcgill university. Proof by contradiction is typically used to prove claims that a certain type. Set theory proof a is a subset of b if and only if a. The theory of numbers provides an excellent source for such examples. That is, it is possible to determine if an object is to be included in the set or not. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen. The empty set is a subset of a, hence it is an element of the power set of a. Set theory for computer science university of cambridge. A set is a collection of objects, which are called elements or members of the set. One way to prove a b is to assume that a is true and b is false. There exist two positive numbers a and b that sum to a negative number. Proof by contradiction example 2 show that 2 is irrational.
But even more, set theory is the milieu in which mathematics takes place today. Alternatively, you can do a proof by contradiction. X for some ordinal then the set z of elements of x. The proof by contradiction is grounded in the fact that any proposition must be either true or.
The cartesian product of two sets is a set, and the elements of that set are ordered pairs. In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. Proof methods mathematical and statistical sciences. Proof by contradiction a proof by contradiction is a proof that works as follows. Basic set theory members of the collection comprising the set are also referred to as elements of the set. Anderson and greg welty what is the relationship between the laws of logic and the existence of god. This is also a set of natural numbers, to which we will apply the wellordered axiom. Proof by contradiction this is an example of proof by contradiction.
It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Please bear in mind that these are only examples of the vast power. Note the difference between the statements such a set does not exist and it is an empty set. Set theory is indivisible from logic where computer science has its roots. To prove that p is true, assume that p is not true. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin.
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