Thus, 3n + 2 is even.

Okay, if a itself is an even number, then a is 2 times some other whole number. Are some square roots rational? Proof by contraposition is a type of proof used in mathematics and is a rule of inference.

If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get: This means that b2 is even, from which follows again that b itself is even. Proof by Contradiction This is an example of proof by contradiction. ¥Use logical reasoning to deduce other facts. Proof by Contradiction (Example 1) •Show that if 3n + 2 is an odd integer, then n is odd. Then we have 3n + 2 is odd, and n is even. So \(p\) is not the largest prime, a contradiction. The structure is simple: assume the statement to be proven is false, and work to show its falsity until the result of that assumption is a contradiction. Discrete Math: A Proof By Contraposition.

And that is a contradiction!!!

Definition 3. We are going to apply the logical rules in proving mathematical theorems. The following simple but wonderful proof is at least as old as Euclid's book The Elements. This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof. More proofs that square root of 2 is irrationalProvided by Cut-the-Knot.org. It does not rely on computers at all, but instead is a "proof by contradiction": if √ 2 WERE a rational number, we'd get a contradiction. This Lecture Now we have learnt the basics in logic. The proof that √ 2 is indeed irrational is usually found in college level math texts, but it isn't that difficult to follow. Still, there seems to be no way to avoid proof by contradiction. Irrational Numbers; Rational Square Roots But this is clearly impossible, since n2 is even. We start with the supposition that the statement is false, and use this assumption to derive a contradiction. A proof by contradiction establishes the truth of a given proposition by the supposition that it is false and the subsequent drawing of a conclusion that is contradictory to something that is proven to be true. a would be odd too. We don't need to know what k is; it won't matter. That is, the supposition that is false followed necessarily by the conclusion from not-, where is false, which implies that is true. We are going to apply the logical rules in proving mathematical theorems. ¥Keep going until we reach our goal. Video 2.

Math 232 - Discrete Math Notes 2.1 Direct Proofs and Counterexamples Axiom: Proposition that is assumed to be true.

Irrational Numbers; Rational Square Roots, More proofs that square root of 2 is irrational, Using a 100-bead abacus in elementary math, Fact families & basic addition/subtraction facts, Add a 2-digit number and a single-digit number mentally, Multiplication concept as repeated addition, Structured drill for multiplication tables, Multiplication Algorithm — Two-Digit Multiplier, Adding unlike fractions 2: Finding the common denominator, Multiply and divide decimals by 10, 100, and 1000, How to calculate a percentage of a number, Four habits of highly effective math teaching. ... 2.2 More Methods of Proof A proof by contradiction establishes that p is true by assuming that p is false and arriving at a contradiction, which is any proposition of the form r ^:r. A proof by contradiction can also be used to prove a statement that is not of the form of an implication. This Lecture Now we have learnt the basics in logic. ¥Keep going until we reach our goal. Discrete Math Lecture 03: Methods of Proof 1. 1. (Attempts to do so have led to the strange world of "constructive mathematics''.) Then n = 2k for some integer k. Then 3n+2 = 3(2k)+2 = 6k +2 = 2(3k +1). One of the most powerful types of proof in mathematics is proof by contradiction or an indirect proof. Therefore √2 cannot be rational. Graphs in Discrete Math: Definition, Types & Uses ... One of several different ways to prove a statement in mathematics is proof by contradiction. We ended at a contradiction; thus our original assumption (that √2 is rational) is not correct. Since n is odd, n = 2k + 1 for some integer k. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Many of the statements we prove have the form P)Q which, when negated, has the form P)˘Q. Proof. Methods of Proof Lecture 3: Sep 9 2. Proof: By contradiction; assume n2 is even but n is odd. Proof by Contradiction 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: something that leads to a contradiction can not be true, and if so, the opposite must be true. A Simple Proof by Contradiction Theorem: If n2 is even, then n is even. 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).

WHY is that a contradiction?

It is powerful because it can be used to prove any statement, in several fields of mathematics. Basic Definitions An integer n is an even number if there exists an integer k such that n = 2k. We have reached a contradiction, so our assumption

In symbols, a = 2k where k is this other number. CS 19: 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.nThese have the following structure: ¥Start with the given fact(s). A proof by contradiction establishes the truth of a given proposition by the supposition that it is false and the subsequent drawing of a conclusion that is contradictory to something that is proven to be true. Sometimes a proof by contradiction can be rewritten as a direct proof. Examples

• Direct proof • Contrapositive • Proof by contradiction • Proof by cases 3. Odd number times odd number is always odd. To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false. This would prove that the statement must be true. How can you tell whether root 10 is a terminating or repeating decimal, or an irrational number?

Thus 3n+2 is even, because it equals 2j for integer j = 3k +1. • Direct proof • Contrapositive • Proof by contradiction • Proof by cases 3. That is, the supposition that P is false followed necessarily by the conclusion Q from not-P, where Q is false, which implies that P is true.

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