Call sa pure set i every such sequence ends with a set s. Indirect proof, also called proof by contradiction, assumes the hypothesis if. Based on the assumption that p is not true, conclude something impossible. Then, the book moves on to standard proof techniques. Perhaps the most obvious thing to say is that there is an epistemological relationship between the two, such that. The sum of two positive numbers is always positive. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. The foundational rules of set theory the laws listed below can be described as the foundational rules of set theory. Chapter 17 proof by contradiction university of illinois. The theory of numbers provides an excellent source for such examples. The proof by contradiction is grounded in the fact that any proposition. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction.
Weve got our proposition, which means our supposition is the opposite. The idea of proof by contradiction is quite ancient, and goes back at least as far as the. R and r has the least upper bound property, then n has a least upper bound. Set theory is indivisible from logic where computer science has its roots. Studies in logic and the foundations of mathematics, volume 102. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Contradiction proofs this proof method is based on the law of the excluded middle. Proof by contradiction is another important proof technique.
In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. Alternatively, you can do a proof by contradiction. Basic concepts of set theory, functions and relations. Please refer to section 10, where the operations on sets are explained in detail.
Propositional logic propositional resolution propositional theorem proving unification today were going to talk about resolution, which is a proof strategy. Let s all sets of type 2 set of all sets not containing themselves as elements s. Proof, sets, and logic department of mathematics boise state. We derive them by going back to the definitions of intersection, union, universal set and empty set, and by considering whether a given element is in, or not in, one or more sets. Assume 0 set theory \a set is a many that allows itself to be thought of as a one.
Chapter 6 proof by contradiction mcgill university. In the foundations of mathematics, russells paradox also known as russells antinomy, discovered by bertrand russell in 1901, showed that some attempted formalizations of the naive set theory created by georg cantor led to a contradiction. Proof is, however, the central tool of mathematics. Russells paradox is the most famous of the logical or settheoretical paradoxes. Set theory an introduction to independence proofs studies. Set theory is a branch of mathematics that investigates sets and their properties. This theorem is proved using cantors first uncountability proof, which differs from the more familiar proof. Prove this proposition using a proof by contradiction.
Discrete mathematicsset theorypage 2 wikibooks, open. The sum of two positive numbers is not always positive. Using this assumption we try to deduce a false result, such as 0 1. In particular, for set theory developed over a certain paraconsistent logic, cantors theorem is unprovable. Axiomatic set theory introduced to control these paradoxes. Cantors first set theory article contains georg cantors first theorems of transfinite set theory, which studies infinite sets and their properties. Proof reasoning by contradiction, assume n is bounded from above. To prove that p is true, assume that p is not true. Russells paradox stanford encyclopedia of philosophy.
In this case, we implicitly assume all of number and set theory to tackle the problem. 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. 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. Introduction since its earliest days, category theory has had to deal with settheoretic questions. First, well look at it in the propositional case, then in the firstorder case. The idea is to assume the hypothesis, then assume the. We take the negation of the given statement and suppose it to be true. Set theory for computer science university of cambridge. Still, there seems to be no way to avoid proof by contradiction.
Also known as the russellzermelo paradox, the paradox arises within naive set theory by considering the set of all sets that are not members of themselves. An introduction to independence proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing. 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. Such a set appears to be a member of itself if and only if it is not a member of itself. 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. One of these theorems is his revolutionary discovery that the set of all real numbers is uncountably, rather than countably, infinite. 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. 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. This text is for a course that is a students formal introduction to tools and methods of proof. 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. 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. Proof methods mathematical and statistical sciences. The treatment of proper forcing in chapter 28 follows jech to a large extent.
That is, it is possible to determine if an object is to be included in the set or not. It will actually take two lectures to get all the way through this. The book starts with the basics of set theory, logic and truth tables, and counting. For example, the set containing only the empty set is a nonempty pure set. Those three methods are direct proof, proof by contradiction, and proof by. The proof that p t in chapter 34 is based upon notes of fremlin. Not sure if this set theory proof attempt involving contradiction is valid. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. Mat231 transition to higher math proofs involving sets fall 2014 8 11. Its a principle that is reminiscent of the philosophy of a certain fictional detective. 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.
Proof by contradiction a proof by contradiction is a proof that works as follows. A good example is freyds special adjoint functor theorem. 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. Proofs and mathematical reasoning university of birmingham. Assume 0 proof of this corollary illustrates an important technique called proof by contradiction. 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. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. So by the axiom of extensionality we have that b0 b. Four basic proof techniques used in mathematics youtube. Proof by contradiction this is an example of proof by contradiction.
The reason is that the proof setup involves assuming. In earlier versions of our logic style manual, we tended to. For pcf in chapters 3032 we follow abraham and magidor. Many of the statements we prove have the form p q which, when negated, has the form p. There exist two positive numbers a and b that sum to a negative number. S contradiction but every element must either be in s or not in s. That is, a proof is a logical argument, not an empir.
If p leads to a contradiction, then p cant be true, and therefore p must be true. In that proof we needed to show that a statement p. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe. See proof 2 is section 5 for a proof by contradiction of n2 is even n is even. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. In mathematics, the notion of a set is a primitive notion. One way to prove a b is to assume that a is true and b is false. Subsets a set a is a subset of a set b iff every element of a is also an element of b.
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. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. A set is pure if all of its members are sets, all members of its members are sets, and so on. The basic concepts of set theory are fairly easy to understand and appear to be selfevident.
Set theory an introduction to independence proofs studies in. Proof by contradiction is typically used to prove claims that a certain type. However, despite its apparent simplicity, set theory turns out to be a very sophisticated subject. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. We will discuss three specific methods of proving theorems that may prove useful to you. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. In other words, you assume that the hypothesis is true but the conclusion is false. A direct proof, or even a proof of the contrapositive, may seem more satisfying. The proof began with the assumption that p was false, that is that. The book first tackles the foundations of set theory and infinitary combinatorics.
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