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Math135Lecture07StudentNotes

# Math135Lecture07StudentNotes - Math 135 Lecture 7 Universal...

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Unformatted text preview: Math 135: Lecture 7: Universal Quantifiers The Choose Method • Choose an “object” in the “universe of discourse” with the “certain” property. This becomes a new statement in the forward process. • Show that, for the chosen object, “something happens”. This becomes a new statement in the backward process. Example 7.1. Coprimeness and Divisibility ( CAD ) If a , b and c are integers and c | ab and gcd( a,c ) = 1, then c | b . is equivalent to Objects: Universe of discourse: Certain property: Something happens: Proof of this statement of CAD: 1. Let a , b and c be integers where c | ab and gcd( a,c ) = 1. 2. By the Extended Euclidean Algorithm and gcd( a,c ) = 1, there exist integers x and y so that ax + cy = 1. 3. Multiplying by b gives abx + cby = b . 4. Since c | ab there exists an integer h so that ch = ab . 5. Substituting ch for ab gives chx + cby = b . 6. Lastly, factoring produces c ( hx + by ) = b . 7. Since hx + by is an integer, c | b . Analysis of Proof For statements that begin with a universal quantifier, we omit the hypoth- esis and conclusion because they are embedded in the components. Objects: Universe of discourse: Certain property: Something happens: Core Proof Technique: 1 Math 135: Lecture 7: Universal Quantifiers Analyzing the Proof of this statement of CAD 1. Let a , b and c be integers where c | ab and gcd( a,c ) = 1. representative “objects” in the “universe of discourse” with the “certain prop- erty”. This becomes a new statement in the forward process....
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Math135Lecture07StudentNotes - Math 135 Lecture 7 Universal...

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