Logic direct proof pdf

Method of direct proof suppose you are trying to prove a universal conditional statement. Oct 28, 2014 proof by contradiction is also known as indirect proof, apago gical ar gument, proof by assuming the opposite, and reductio ad imp ossibility. Four basic proof techniques used in mathematics youtube. A direct proof is a method of showing whether a conditional statement is true or false using known facts and rules. Then i will present a large number of arguments by logic that can be used in proofs, i will also give an example of a. Contents preface vii introduction viii i fundamentals 1. In order to directly prove a conditional statement of the form if p, then q, it suffices to. One of the most important proof techniques is the method of direct proof. Therefore, alice is either a math major or a csi major. To move through indirect proof logic, you need real confidence and deep content knowledge.

Winkler introduction in what follows, i will begin by giving an overview of logic. A true mathematical statement whose truth can be verified. The book starts with the basics of set theory, logic and truth tables, and counting. Our objective is to reduce the process of mathematical reasoning, i. A true mathematical statement whose truth is accepted without proof. Simon singh a proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true. All major mathematical results you have considered. Be able to use the resolution proof system in proposiitonal logic and in predicate logic. That same idea of indenting to indicate that were making an assumptionis used in another very useful strategy for writing formal proofs, one known as indirect proof. Direct proof and counterexample 1 in this chapter, we introduce the notion of proof in mathematics.

Paris is in france true, london is in denmark false, 2 direct proof is a sequence of statements which are either givens or deductions from previous statements, and whose last statement is the conclusion to be proved. Between the first and last line we use logic, definitions and. Before we explore and study logic, let us start by spending some time motivating this topic. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Direct proof assumes a given hypothesis, or any other known statement, and. Assume p, and then use the rules of inference, axioms, defi nitions, and logical equivalences to prove q. The proof of a proposition is an argument that will convince any reader. Be able to express and formalize in a logical language properties of models such as graphs, strings and transition systems, and be able to determine the truth or falsity of such formulas in a given model. Proof techniques jessica su november 12, 2016 1 proof techniques here we will learn to prove universal mathematical statements, like \the square of any odd number is odd.

Proof by contradiction a proof by contradiction is a proof that works as follows. Then i will present a large number of arguments by logic that can be used in proofs, i will also give an example of a proof by truth tables and a more traditional proof. A direct proof assumes that the hypothesis of a conjecture is true, and then uses a series of logical deductions to prove that the conclusion of the conjecture is true. Use rules of inference, axioms, and logical equivalences to show that q must also be true. The following simple but wonderful proof is at least as old as euclids book the elements. A rule of inference is a logical rule that is used to deduce one statement from others. Samvel atayan and brent hickman august 11, 2009 additional readings. To be considered a theorem as opposed to a result, the statement should be particularly significant or interesting. Mathematicians define terms very carefully and precisely and consider it important to learn definitions.

Propositions a proposition is a declarative sentence that is either true or false but not both. The mathematical proof is really to show that q 1 q 2 q k. 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. Each step of the argument follows the laws of logic. One way to prove a b is to assume that a is true and b is false. An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. You must know the meanings of all terms that occur in the statement. In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. Its easy enough to show that this is true in speci c cases for example, 3 2 9, which is an odd number, and 5 25, which is another odd number. This has a very old lineage, being known in medieval times as reductio ad absurdum, which means showing that a position leads to an absurdity. The three steps seem simple, much as a onepage cartoon diagram makes assembling furniture seem simple. Theorem that is not interesting on its own, but helps prove another theorem.

In mathematics, a statement is not accepted as valid or correct unless it is accompanied by a proof. Logic and proof mary radcli e in this set of notes, we explore basic proof techniques, and how they can be understood by a grounding in propositional logic. Paris is in france true, london is in denmark false, 2 oct 28, 2014. We will show how to use these proof techniques with simple examples, and demonstrate that they work using truth tables and other logical tools. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion is true under the assumption that the premisses are true. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. Euclidean algorithm to nd the gcd lets use the euclidean algorithm to nd gcd38. The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics. Conditional and indirect proof introduction to logic. Two direct proofs of lehmussteiners theorem are proposed. A direct proof, or even a proof of the contrapositive, may seem more satisfying. In other words, you assume that the hypothesis is true but the conclusion. Suppose n is any particular but arbitrarily chosen even integer. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile.

Proofs in propositional logic propositions and types like in many programming languages, connectors have precedence and associativity conventions. It is a particular kind of the more general form of argument known as. Their improper use results in unclear and even incorrect arguments. Fundamentals of logic and techniques for mathematical proofs. Proof by contradiction this is an example of proof by contradiction. 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. To prove that p is true, assume that p is not true. Still, there seems to be no way to avoid proof by contradiction. Language, proof and logic second edition dave barkerplummer, jon barwise and john etchemendy in collaboration with albert liu, michael murray and emma pease.

Mat231 transition to higher math direct proof fall 2014 14 24. Direct proof example 2 show that if m and n are both square numbers, then m n is also a square number. The possible truth values of a statement are often given in a table, called a truth table. Many of the statements we prove have the form p q which, when negated, has the form p. A proof is an argument from hypotheses assumptions to a conclusion. Direct proofs at this point, we have seen a few examples of mathematicalproofs. The simplest way to prove a b is to assume a the \hypothesis and prove b the \conclusion. Then, the book moves on to standard proof techniques. This implies that there are integers u and v such that m u2 and n v2. Negation 3 we have seen that p and q are statements, where p has truth value t and q has truth value f. Attempts to do so have led to the strange world of constructive mathematics. The proper use of variables in an argument is critical. Introduction definitions in order to evaluate the truth or falsity of a statement, you must understand what the statement is about.

Direct proof mccpdobson0211 introduction a directproofis one of the most familiar forms of proof. We will show how to use these proof techniques with simple examples, and demonstrate that they. In order to directly prove a conditional statement of the form if p, then q, it suffices to consider the situations in which the statement p is true. From firstorder logic we know that the implication p. The page will try to find either a countermodel or a tree proof a. Studies in logic and the foundations of mathematics. A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems. Intro rules of inference proof methods rules of inference for propositional logic which rule of inference is used in each argument below.

The method of the proof is to takes an original statement p, which we assume to be true, and use it to show directly that another. Then the statement we wish to prove can be interpreted as p. The truth values for two statements p and q are given in figure 1. We use it to prove statements of the form if p then q or p implies q which we can write as p. See proof 2 in section 5 for a direct proof of n is even n2 is even. Logical models illustrate the essence of specific types of indirect proofs. Proofs and mathematical reasoning university of birmingham. Give a direct proof of the theorem if n is an odd integer, then n2 is odd.