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onenew - Introduction to Discrete Structures One....

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Introduction to Discrete Structures One. Introduction and Overview 1.1. Notations and Terminology Formal systems or mathematics in general use their own notations and language to convey the ideas and conduct formal reasoning. Some examples are as follows: If n is an integer divisible by 4, then n is an even integer. (Use if-then for an implication relationship.) If A and B are two sets, then A B A B , which means (in plain words) the intersection of sets A and B is contained in the union of A and B . (Use the set theoretic notations in an implication.) Prove that A for any set A. To prove that A , we need to prove that if x , then x A , by the definition of " ". Since x is false, because the empty set contains no elements; thus, the implication x x A is vacuously true, so the proof is completed. (Use logical reasoning in a proof .) 1-1, ©Dr. S. Lang
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1.2. Proof Methods There are essentially two methods in conducting formal proofs: the direct method and the indirect method. The following are their descriptions and some examples illustrating the ideas. 1.2.1. Direct Proofs A mathematical (formal) statement is proved by starting with its assumptions, and by using the definitions and other known facts (theorems), the conclusion is reached via a sequence of steps based on logical reasoning, i.e., each step is supported by definition, theorem, or logic. Many (or most) theorems are proved by the direct methods. For example, we will prove the following two theorems concerning integers, using the direct method: The sum of two odd integers is even. If a is a divisor of b , and b is a divisor of c , then a is a divisor of c . We need to use the relevant definitions, notations, and known facts (theorems) to prove the above two theorems. 1-2, ©Dr. S. Lang
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First, let us recall some related definitions and theorems in the domain of integers. The set of integers, Z , contains numbers 0, 1, +1, 2, +2, -3, +3, etc. That is, Z = { …, 2, 1, 0, 1, 2, … }. The set of integers and the three operations + (add), – (subtract), and • (multiply), satisfy the following properties: (Closure Property under +, –, •) If a , b are integers, then a + b, a b, a b (or simply ab ) are integers. (Commutative Law) If a , b are integers, then a + b = b + a , a b = b a . (Associative Law) If a , b, c are integers, then ( a + b ) + c = a + ( b + c ), and ( a b ) c = a ( b c ). (Distributive Law) If a , b, c are integers, then a ( b + c ) = a b + a c . (Identity Elements) For all integer a , a + 0 = 0 + a = a ; a • 1 = 1 • a = a . (Additive Inverse) For all integer a , a + (– a ) = (– a ) + a = 0. Definition (Divisibility) If a , b are integers, then a | b , or a is a divisor of b , if a 0 and there exists an integer c such that b = a c. Definition (Even and odd) An integer a is even if there exists an integer b such that a = 2 b (or, equivalently, if 2 | a ); an integer a is odd if there exists an integer b such that a = 2 b + 1.
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onenew - Introduction to Discrete Structures One....

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