Theorem In Discrete Structure

Discrete Structures Lecture Notes for CSE 191 Matthew G. Knepley Feng-Mao Tsai Department of Computer Science and Engineering University At Bu alo

Discrete Structure CS-302 B.Tech RGPV notes AICTE flexible curricula Bachelor of technology and onto function, inverse function, composition of functions, recursively defined functions, pigeonhole principle. Theorem proving Techniques Mathematical induction, Proof by contradiction. UNIT 2 Algebraic Structures Definition, Properties

notation or prove some theorem in class, you can use these freely in your homework and exams, provided that you clearly cite the appropriate theorems. In writing and speaking mathematics, a delicate balance is maintained between being formal and not getting bogged down in minutia.1 This balance usually becomes second-nature with experience.

Recall that we cannot use the Master Theorem if fn, the non-recursive cost, is not a polynomial. There is a limited 4th condition of the Master Theorem that allows us to consider polylogarithmic functions. Corollary If for some k 0 then This final condition is fairly limited and we present it

We now present a proof of the Pythagorean Theorem. b a c Figure 4.2 A right triangle with sides of length a, b, and c Proposition 4.2 Pythagorean Theorem. In a right triangle where the hypotenuse has length c and the other two sides have lengths a and b, we have a2 b2 c2. Proof. Consider a right triangle like the one in Figure 4.2.

The Fundamental Theorem of Arithmetic Theorem 1 Every positive integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size. Intro to Discrete StructuresLecture 12 - p. 1339

Moreover, this structure of using quotlet 92x92 bequot is an implicit universal quantifer. For all 92x92 which satisfy this property, the inference theorem holds. With a little practice, we can extract the impliciation from such statements. Indeed, the quotlanguage of mathematicsquot is itself a language, and we need to understand it through

Theorem If x is a positive integer, then x x2. What is meant is For allpositive integers x x2. Many theorems are universal statements. And some theorems can be universal statements over multiple variables with different domains TheoremIf x and y are positive real numbers and n is a positive integer, then x yn xn yn.

Definition A theorem is a statement that can be shown to be true. We demonstrate that a theorem is true with a proof valid argument using Definitions integers, the real numbers, or some of the discrete structures that we will study in this class. Often the universal quantifier needed for a precise statement of a theorem is

- Proving one part of thefundamental theorem of arithmetic. Theorem Any integer gt 1 can be written as a product of one or more primes Proof by contradictionusing WOP! Let S be the set of all integers greater than 1 that cannot be written as a product of one or more primes. If S is non-empty, there is a least element in it by WOP.