A statement or proposition is an assertion which is either true or. Proof by contradiction a proof by contradiction is a proof that works as follows. It can be a calculation, a verbal argument, or a combination of both. A contradiction is a proposition that is always false. These words have very precise meanings in mathematics which can di.
Wuct121 logic tutorial exercises solutions 2 section 1. What else is there that logicians are doing that isnt considered part of mathematical logic. Mathematical proofmethods of proofproof by contradiction. This is an important technique for proving mathematical results. These include proof by mathematical induction, proof by contradiction, proof by exhaustion, proof by enumeration, and many. Since q2 is an integer and p2 2q2, we have that p2 is even.
To prove that p is true, assume that p is not true. Based on the assumption that p is not true, conclude something impossible. Most of the steps of a mathematical proof are applications of the elementary rules of logic. Wuct121 discrete mathematics logic tutorial exercises. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. A compound statement s is called a contradiction if the truth value of s is false. Mat231 transition to higher math proof by contradiction fall 2014 6 12. The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics. One question worth 4 marks is asked from this topic in jeemains paper. Adding sets and quanti ers to this yields firstorder logic, which is the language of modern mathematics. Arguments in propositional logic a argument in propositional logic is a sequence of propositions. It is important to realize that in mathematics, until an idea is applied to something concrete ite no meaning. Logic and proof to begin with, i will need to present the basic method of formal mathematics. I will try to make this as basic as possible and still be useful.
Logic alphabet, a suggested set of logical symbols mathematical operators and symbols in unicode polish notation list of mathematical symbols notes 1. Propositional logic, truth tables, and predicate logic rosen, sections 1. The argument is valid if the premises imply the conclusion. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic. In mathematics, a contradiction is the assertion of a statement and its negation, or equivalently, a statement that can never be true. Although this character is available in latex, the mediawiki tex system doesnt support this character. The study of logic helps in increasing ones ability of systematic and logical reasoning. In comparison to computational math problems, proof writing requires greater emphasis on mathematical rigor, organization, and communication. It cant be both rational and irrational, so theres our contradiction. Logic is the subject that deals with the method of reasoning. A contradiction is equivalent to the negation of a tautology. Introduction to logic and set theory 202014 bgu math.
Every statement in propositional logic consists of propositional variables combined via logical connectives. Methods of reasoning, provides rules and techniques to determine whether an argument is valid theorem. Furthermore, many would maintain that the concept of god must conform to the laws. Digital electronic circuits are made from large collections of logic gates, which are. In general, to prove a proposition p by contradiction, we assume that p is false. Four basic proof techniques used in mathematics youtube. Generally, students dont pay much attention to this topic especially those who are targeted for jeeadvanced. In this introductory chapter we deal with the basics of formalizing such proofs. Indeed, remarkable results such as the fundamental theorem of arithmetic can be proved by contradiction e. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematical proofs. A compound proposition is satisfiable if there is at least one assignment of truth values to the variables that makes the statement true.
It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. 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. We will prove this statement using a proof by contradiction. 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 word tautology is derived from a greek word where tauto means same and logy means logic. It is a particular kind of the more general form of argument known as. An argument form is an argument that is valid no matter what propositions are substituted into its propositional variables. Problems on discrete mathematics1 ltex at january 11, 2007. Proving conditional statements by contradiction 107 since x. By grammar, i mean that there are certain commonsense principles of logic, or proof techniques, which you can. Variables and connectives propositional logic is a formal mathematical system whose syntax is rigidly specified.
The basic idea for a proof by contradiction of a proposition is to assume the proposition is false and show that this leads to a contradiction. Hence, there has to be proper reasoning in every mathematical proof. If maria learns discrete mathematics, then she will find a good job. Tautology in math definition, logic, truth table and examples. But we havent yet proved p to be true, so the contradiction is not obvious. The system we pick for the representation of proofs is gentzens natural deduction, from 8. The study of logic helps in increasing ones ability of. Propositional logic, truth tables, and predicate logic. Statement proposition a statement is an assertive sentence which is either true or false but not both a true statement is called valid statement. Logic the main subject of mathematical logic is mathematical proof. Broadly speaking, there are two ways that one can show. A compound statement is a tautology if it is true re gardless of the truth values. Illustrating a general tendency in applied logic, aristotles law of noncontradiction states that it is impossible that the same thing.
A compound statement is made with two more simple statements by using some conditional words such as and, or, not, if, then, and if and only if. The following theory of symbolic logic recommended itself to me in the first instance by its ability to solve certain contradictions, of which the one best known to mathematicians is buralifortis concerning the greatest ordinal. So here im trying to make it easily covered through this note. New to proving mathematical statements and theorem. Without loss of generality we can assume that a and b have no factors in common i. Mathematical reasoning for jeemains sandeep bhardwaj. A compound proposition that is neither a tautology or a contradiction is called a. This method of proof is also one of the oldest types of proof early greek mathematicians developed. Mathematical reasoning is a topic covered under the syllabus of jeemains only, excluding jeeadvanced exam. A tautology is a compound statement which is true for every value of the individual statements. 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.
Discrete mathematics logic tutorial exercises solutions 1. The vocabulary includes logical words such as or, if, etc. A primer on mathematical proof a proof is an argument to convince your audience that a mathematical statement is true. A mathematical proof of a statement strongly depends on who the proof is written for. Propositional logic is a mathematical system for reasoning about. Ive studied mathematics at uni with a few logic courses tailored to mathematicians, and i guess i have only been in contact with the mathematical one. Propositional logic propositional logic is a mathematical system for reasoning about propositions and how they relate to one another. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. Mathematical logic introduction mathematics is an exact science.
Before we explore and study logic, let us start by spending some time motivating this topic. The opposite of a tautology is a contradiction or a fallacy, which is always false. To have a uent conversation, however, a lot of work still needs to be done. A primer on mathematical proof university of michigan. Proof methods mathematical and statistical sciences. No matter what the individual parts are, the result is a true statement. Contradiction proofs this proof method is based on the law of the excluded middle. A tautology in math and logic is a compound statement premise and conclusion that always produces truth. Proof by contradiction in logic and mathematics is a proof that determines the truth of a statement by assuming the proposition is false, then working to show its falsity until the result of that assumption is a contradiction. In studying mathematical logic we shall not be concerned with the truth value of. 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. Propositional logic enables us to formally encode how the truth of various propositions influences the truth of other propositions. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile.
Since a rational number where a and b are integers and b. 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. We start with the language of propositional logic, where the rules for proofs are very straightforward. From the name mathematical logic i get the impression that it is an area of logic. Then we argue that, if this is the case, we can lead to a result showing. It provides us rules for determining the validity of a given argument in proving theorem. Every statement in propositional logic consists of propositional. The metaphor of a toolbox only takes you so far in mathematics. In fact, the apt reader might have already noticed that both the constructive method and contrapositive method can be derived from that of contradiction. Mathematical logic as based on the theory of ty pes. Determine if certain combinations of propositions are.
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