Math135Lecture03-StudentNotes

Math135Lecture03-StudentNotes - Math 135: Lecture 3:...

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Unformatted text preview: Math 135: Lecture 3: Forwards and Backwards Definition 3.1. An integer m divides an integer n , and we write m | n , if there exists an integer k so that n = km . Example 3.2. Divisibility (a) 3 | 6 (b) 5- 6 (c) a | (d) For all non-zero integers a , 0- a since Proposition 3.3 (Transitivity of Divisibility TD ) . Let a,b,c Z . If a | b and b | c , then a | c . Proof of TD: 1. Since a | b , there exists an integer r so that b = ra = b . 2. Since b | c , there exists an integer s so that c = sb = c . 3. Substituting ra for b in the previous equation, we get c = sb = s ( ra ) = ( sr ) a . 4. Since sr is an integer, a | c . Reading a Proof 1. Explicitly identify the and the . 2. Explicitly identify the . 3. Record any , usually definitions or previous knowl- edge. 4. each step with reference to the definitions, previous knowledge or techniques used. 5. where necessary and justify these steps with reference to the definitions, previous knowledge or techniques used. Analysis of Transitivity of Divisibility (TD) Let a,b,c Z . If a | b and b | c , then a | c . Hypothesis: Conclusion: Core Proof Technique: Preliminary Material: 1 Math 135: Lecture 3: Forwards and Backwards Example 3.4. Analysis of Proof of TD . Proof of TD: 1. Since a | b , there exists an integer r so that ra = b . 2. Since b | c , there exists an integer s so that sb = c ....
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Math135Lecture03-StudentNotes - Math 135: Lecture 3:...

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