chap1defs - Part I. Integers: (Closure Property under +, ,...

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Part I. Integers: (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 . (Law of Identity Elements) For all integer a , a + 0 = 0 + a = a ; a • 1 = 1 • a = a . (Law of 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. Theorem. The sum of two odd integers is an even integer. Theorem. If a | b , and b | c , then a | c , where a , b , and c are integers. Definition (Prime Number) A positive integer n > 1 is a prime number if its divisor must be either 1 or n itself (that is, if m | n , then m = 1 or m = n ). Theorem (Fundamental Theorem of Arithmetic) Any integer n > 1 can be written as a product of prime numbers. Further, this product is unique except for rearrangement of the terms. Theorem (Euclid’s Division Theorem) For any integers n and m , m 1, there exist integers q and r , 0 r < m , such that n = m q + r . Further, these integers q and r are unique. Theorem. Every integer n is either even or odd. Theorem. If a , b are two integers and a b is even, then at least one of a and b is even. Theorem (The Pigeonhole Principle) If n + 1 or more pigeons are nested in n holes, n 1, then at least one hole nests more than one pigeon. Theorem. The number 2 (the square root of 2) is not a fraction of two integers (i.e., not a rational number). Theorem.
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chap1defs - Part I. Integers: (Closure Property under +, ,...

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