Proportional logic proof by contradiction pdf

The big idea in propositional calculus is to study the structure of simple state ments. Therefore2 name abbreviation rule comments modus ponens mp p e q p \ q pithy statement. The sum of two positive numbers is always positive. In this module, we will consider propositional logic, which will look familiar to you from. Perhaps students at the postsecondary level find proof so difficult because their only experience in writing proofs has been in a. Discrete mathematics propositional logic tutorialspoint. Proof by contradiction this is an example of proof by contradiction. A proposition is a declarative statement which is either true or false. Seem 5750 7 propositional logic a tautology is a compound statement that is always true. To prove that p is true, assume that p is not true. Since by the law of bivalence a proposition must be either true or false, and its falsity has been shown impossible, the.

Proof is a very difficult area for undergraduate mathematics students. Some statements cannot be expressed in propositional logic, such as. Brouwe r claimed that proof by contradiction was sometimes invalid. This topic has a huge history of philosophic conflict. A proposition is the basic building block of logic. 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.

Solutions to propositional logic proof exercises october 6, 2016 1 exercises 1. If a proposition is true, then we say its truth value is true, and if a proposition is false, we say its truth value is false. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. Mathematical proofmethods of proofproof by contradiction. In general, to prove a proposition p by contradiction, we assume that p is false. Proofs in proposition logic and predicate logic labri. In that proof we needed to show that a statement p.

In fact, the apt reader might have already noticed that both the constructive method and contrapositive method can be derived from that of contradiction. Before we explore and study logic, let us start by spending some time motivating this topic. Propositional logic, truth tables, and predicate logic rosen, sections 1. Chapter 6 proof by contradiction mcgill university.

Certain tautologies of propositional logic allow us to explain such common proof techniques as proof by contradiction or proof by contrapositive section. Any formal system can be considered a logic if it has. In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or. All but the nal proposition are called premises and the nal proposition is called the conclusion. Discrete mathematics introduction to propositional logic. Propositional logic richard mayr university of edinburgh, uk richard mayr university of edinburgh, uk discrete mathematics. Please answer each question completely, and show all of your work. Propositional logic cheriton school of computer science. We show that the extended all,, remains socratically complete thus. Reasoning and proof cannot simply be taught in a single unit on logic, for example, or by doing proofs in geometry. Many of the statements we prove have the form p q which, when negated, has the form p. From this assumption, p 2 can be written in terms of a b, where a and b have no common factor. Propositional logic is concerned with statements to which the truth values, true and false, can be assigned.

To prove p, assume p and derive a contradiction such as p. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. Nov 27, 2016 propositional logic first order predicate logic propositional calculus gate net part 1. If we were formally proving by contradiction that sally had paid her ticket, we would assume that she did not pay her ticket and deduce that therefore she should have got a nasty letter from the council. In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the propositions being false would imply a contradiction. Angelo, bruno and carlo are three students that took the logic exam. Types of propositions atomic proposition and compound proposition. A primer for logic and proof appalachian state university. Use the truth tables method to determine whether the formula. Prove that if you pick 22 days from the calendar, at least. Beginning around 1920, a prominent dutch mathematician by the name of l. There are some issues with this example, both historical and pedagogical.

Its a principle that is reminiscent of the philosophy of a certain fictional detective. There exist two positive numbers a and b that sum to a negative number. The purpose is to analyze these statements either individually or in a composite manner. Then, you negate the desired conclusion so you have to say what youre trying to prove, but what were going to do is essentially a proof by contradiction. Chapter 17 proof by contradiction university of illinois. In this paper, we extend previous work to include negation, disjunction, and the ability to make assump tions and reason by contradiction. Propositional logic internet encyclopedia of philosophy. A proposition is a collection of declarative statements that has either a truth value true or a. Justify all of your decisions as clearly as possible. Propositional logic first order predicate logic propositional calculus gate net part 1.

Predicate logic can express these statements and make inferences on them. Noncontradiction is also a theorem in propositional logic. So this is a valuable technique which you should use sparingly. The more work you show the easier it will be to assign partial credit. Methods of proof for boolean logic proof by contradiction. Classify each of the following formulas as a tautology, a contradiction, or a. First, well look at it in the propositional case, then in the firstorder case. Weve got our proposition, which means our supposition is the opposite. We can combine resolution with proof by contradiction where we assert the negation of what we wish to prove, and from that premise derive false to direct our search towards smaller and smaller clauses, with the goal of producing false. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect.

Logical proof is proof that is derived explicitly from its premises without exception. Mathematics introduction to propositional logic set 1. If our supposition in a proof by contradiction was there exists some integer n such that the product of n and its. To clarify the manipulations we perform in logical proofs, we will represent. This states that an assertion or mathematical statement cannot be both true and false.

Based on the assumption that p is not true, conclude something impossible. The proof began with the assumption that p was false, that is that. It will actually take two lectures to get all the way through this. We can prove a is not true by finding a counter example. Gradeup gate, ese, psus exam preparation 18,282 views. Recall that a number is rational if it equals a fraction of two integers, and it is irrational if it cannot be. Propositional logic, truth tables, and predicate logic rosen. For many students, the method of proof by contradiction is a tremendous gift and a trojan horse, both of which follow from how strong the method is. In more recent times, this algebra, like many algebras, has proved useful as a design tool. Home geometry logic and proof topics types of proofs. Far too many authors of contemporary texts in informal logic keeping an eye on the sorts of arguments found in books on formal logic forget, or underplay, how much of our daily reasoning is concerned not with arguments leading to truthvalued conclusions but. For example, chapter shows how propositional logic can be used in computer circuit design.

In order to avoid to prove contradictions, there is no introduction rule nor. Find the converse, inverse, and contrapositive of it is raining is a suf. For every action, there is an equal and opposite reaction. Propositional logic in this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to aristotle, was to model reasoning. The early emphasis on formal logic proofs distinguishes this book from many. Why cant we use one counterexample as the contradiction to the contradicting statement. The simple form of logic is propositional logic, also called boolean logic. Alternatively, you can do a proof by contradiction. The truth value of a proposition is true denoted as t if it is a true statement, and false denoted as f if it is a false statement. There is no integer solution to the equation x 2 5 0. A real proof by contradiction will arrive at some kind of internal contradiction. Lets consider a propositional language where aaldo passed the exam, bbruno passed the exam, ccarlo passed the exam.

It is a particular kind of the more general form of argument known as. It is defined as a declarative sentence that is either true or false, but not both. These include proof by mathematical induction, proof by contradiction, proof by exhaustion, proof by enumeration, and many. It is a technique of knowledge representation in logical and mathematical form. The current context contains two mutually contradictory propositions. Here are some good examples of proof by contradiction. An introduction to proof by contradiction, a powerful method of mathematical proof. A compound proposition is satisfiable if there is at least one assignment of truth values to the variables that makes the statement true. Propositional logic propositions examples gate vidyalay. Proof by contradiction proving something false there is a very important method for proving something false proof by contradiction a. Propositional logic, truth tables, and predicate logic. We now show how logic is used to represent knowledge. The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics.

The sum of two positive numbers is not always positive. In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Our next example follows their logic to prove that 2 is irrational. Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining andor modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. A proposition or statement is a sentence which is either true or false. Propositional logic pl is the simplest form of logic where all the statements are made by propositions. Here is an example of an ordinary proof, in contemporary. Propositional logic in artificial intelligence javatpoint. Propositional logic propositional resolution propositional theorem proving unification today were going to talk about resolution, which is a proof strategy. Most of the steps of a mathematical proof are applications of the elementary rules of logic. Since we have shown that p f is true, it follows that the contrapositive tp also holds. In formal logic, a valid argument is an argument that is structured in such a way that if all its premises ar. Rules of inference, propositional logic1 keith burgessjackson 9 september 2017 implication rules \ df.