Rather we have started to get acquainted with the process of rigorously proving

Rather we have started to get acquainted with the

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3 IMPORTAglyph1197T In a proof, we are given some information, and logic is employed to effect deduction, that is, to make new conclusions from the given information. In our example, we see that given two facts: Definition 1.1 and 8=2∙4 , it follows “logically” that 8 is even. So there are two essential ingredients in a proof: inputted information and logic . IMPORTAglyph1197T Definitions represents a major input in a proof. There is more than one way to define something, and our proof depends on the definitions we use . Example 1.3 We give an alternative definition of even integers, and a corresponding proof that 8 is even. Definition 1.1a An integer ݔ is even if ݔ/2 is an integer. Proof Since 8/2=4 , which is an integer, we have 8 is even by Definition 1.1a. Example 1.4 Similarly, we can disprove that 7 is even (or equivalently, to prove that 7 is not even). Proof 1 Since 7 is between 6=2ሺ3ሻ and 8=2ሺ4ሻ , and there is no integer between 2 and 3 , we know that there is no integer ݊ such that 2݊=7 . So, by Definition 1.1, 7 is not even. Proof 2 Since 7/2=3.5 , which is not an integer, 7 is not even by Definition 1.1a. Example 1.5 How can we define an odd integer? Definition 1.2 An integer ݔ is odd if ݔ=2݊+1 for some integer ݊ . Definition 1.2a An integer ݔ is odd if ݔ is not even. IMPOTAglyph1197T Definition 1.2a is legitimate because we have already defined an even integer. Example 1.6 Prove that 7 is an odd number. Proof 1 Since 7=2ሺ3ሻ+1 and 3 is an integer, by Definition 1.2, 7 is an odd number. Proof 2 Since we have proved that 7 is not even, by Definition 1.2a, 7 is odd. In Proof 2, we use a previously established fact, namely, 7 is not even (Example 1.4). IMPOTAglyph1197T Apart from using definitions, we often use proven theorems or results in proofs. We move beyond “even” and “odd”, and define another fundamental mathematical concept: divisibility . Definition 1.3 An integer ܽ divides an integer ܾ (denoted ܽ|ܾ ) if ܾ=ܽ݊ for some integer ݊ . Definition 1.4 An integer ܽ is a divisor ( factor ) of an integer ܾ if ܽ divides ܾ . Definition 1.5 An integer ܾ is a multiple of an integer ܽ if ܽ is a divisor of ܾ . IMPOTAglyph1197T ܽ divides ܾ ” is equivalent to “ ܽ is a divisor of ܾ ”, and “ ܾ is a multiple of ܽ ”. We introduce a notation : ܽ|ܾ . Notations are extremely important in mathematics! We use ܽ∤ܾ to denote “ ܽ does not divide ܾ ”, just as denotes inequality. In general, “slash” means “not” . Example 1.7 Suppose we have the following alternative definition of divisibility. Definition 1.3a An integer ܽ divides an integer ܾ (denoted ܽ|ܾ ) if ܾ/ܽ is an integer.
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  • Fall '13
  • RyanKinser
  • Elementary arithmetic, Natural number, Prime number, Even and odd functions, Parity, Evenness of zero

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