Use P to show that Q must be true. MAT231 (Transition to Higher Math) Direct Proof Fall 2014 14 / 24. truth or falsehood of a given statement by a straightforward combination of n = 2k for some integer k. Multiply both sides by −1, we get The earliest use of proofs was prominent in legal proceedings. Each of the triangles has sides a and b and hypotenuse c. The area of a square is defined as the square of the length of its sides - in this case, (a + b)2. Your proof is correct. A direct proof is a method of showing whether a conditional statement is true or false using known facts and rules. It follows that x + y has 2 as a factor and therefore is even, so the sum of any two even integers is even. – Logical deductions or inference rules are used to prove new – A proof is a sequence of logical deductions from axioms and {\displaystyle {\frac {1}{2}}ab. Now customize the name of a clipboard to store your clips. }, We know that the area of the large square is also equal to the sum of the areas of the triangles, plus the area of the small square, and thus the area of the large square equals Proof. to make a series of deductions that eventually prove the conclusion of the conjecture to be true 4. Proof as we know it came about with one specific question: “what is a proof?” Traditionally, a proof is a platform which convinces someone beyond reasonable doubt that a statement is mathematically true. The area of a triangle is equal to Because a and b are Most simple proofs are of this kind. . }, Removing the ab that appears on both sides gives, By definition, if n is an odd integer, it can be expressed as, Since 2k2+ 2k is an integer, n2 is also odd. Example 1 (Version I): Prove the following universal statement: The negative of any even integer is even. Method of direct proof 1. We know that the area of the large square is equal to (a + b)2. The type of logic employed is almost invariably first-order logic Introduction To Proofs Discrete Mathematics, Discrete Math Lecture 03: Methods of Proof, Chapter-3: DIRECT PROOF AND PROOF BY CONTRAPOSITIVE, No public clipboards found for this slide. Direct proof methods include proof by exhaustion and proof by induction. This is the “simplest” method and sometimes it can seem that the proof … Direct Proof In a direct proof one starts with the premise (hypothesis) and proceed directly to the conclusion with a chain of implications. You can change your ad preferences anytime. Discussion If a direct proof of an assertion appears problematic, the next most natural strat- egy to try is a proof of the contrapositive. 2 Examples of Direct Method of Proof . Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Euclidean Algorithm to nd the GCD Lets use the Euclidean Algorithm to nd gcd(38;22). {\displaystyle 4({\frac {1}{2}}ab)+c^{2}. A 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. Theorem 1. Method of Direct Proof. It's basically if p, then q . Common proof rules used are modus ponens and universal instantiation.[2]. Learn more. The earliest form of mathematics was phenomenological. On occasion, analogical arguments took place, or even by “invoking the gods”. ¥Use logical reasoning to deduce other facts. Assume that a and b are consecutive integers. Direct Proofs At this point, we have seen a few examples of mathematical)proofs.nThese have the following structure: ¥Start with the given fact(s). Looks like you’ve clipped this slide to already. The word ‘proof’ comes from the Latin word probare,[3] which means “to test”. [4] This led to a natural curiosity with regards to geometry and trigonometry – particularly triangles and rectangles. An integer n is even iff there exists an integer s so that n = 2s. 2 The type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. 2. In my first post on my journey for improving my mathematical rigour I said that I would go through a few different techniques for conducting proofs. Consider two even integers x and y. Does "proof by direct method" have some technical meaning given by your teacher? proposition in question. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. By definition of even number, we have. ¥Keep going until we reach our goal. Assume the hypothesis to be true 3. 1 This statement has the form p! The first one I want to dabble into is direct proofs. However, the area of the large square can also be expressed as the sum of the areas of its components. . In a little more technical language we say that the underlying idea of direct proof is to show that every element of a domain satisfy a certain property and we accomplish this task as follows: 1. If so, that might be why the proof was deemed wrong. There are only two steps to a direct proof (the second step is, of course, the tricky part): 1. – An axiom is a proposition that is simply accepted as true. a 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.