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# s.inc - that avoids words “ancestor” “descendant”...

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Numbers & Polys MAS3300 3244 Exam-S Prof. JLF King 7Nov2005 Note. This is an open brain, open ( pristine ) Sig- monNotes exam. Please write each solution on a separate sheet of paper. Please be sure to write ex- pressions unambiguously e.g, the expression “1 /a + b should be bracketed either [1 /a ] + b or 1 / [ a + b ]. Be careful with negative signs! Every if ” must be matched by a “ then .” S1: Compute D := gcd(51 , 30)= . . . via the Lightning Bolt ( Euclidean ) algorithm. Give inte- gers S = . . . and T = . . . so that 51 S +30 T = D . S2: For each of the following statements in quotes, circle one of T F . Then provide a proof or a CEX with explicit numbers . a “Addition distributes over mult.” T F b On R define x / y := h x · y i + y . Then “binop / is associative”. T F S3: Please prove Thm1.4d: ( P.2 ) If e R is an multiplicative -identity then e = 1 . S4: A person P is a prewash if P is a female ancestor of George Washing- ton. Write a recursive definition that starts Prewash : A person who. . . that avoids words “ancestor”, “descendant” etc.
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Unformatted text preview: . . that avoids words “ancestor”, “descendant” etc. It may use “child”, “parent”, “son”, “mother” etc. S5: Let “ * ” mean “ theorems earlier than (1.15f)”. Using ( * ) prove: Lemma: If z 6 = 0 then z 2 is posi-tive. Now use this and ( * ) to prove that 1 < 0. Bonus: Write the set of Primes using set-builder notation. S1: 65pts S2: 50pts S3: 55pts S4: 55pts S5: 50pts Bonus: 15pts Total: 275pts Ordinal: Print name ........................................ Honor Code: “I have neither requested nor re-ceived help on this exam other than from my professor (or his colleague).” Signature: ..................................... Filename: Classwork/NaPo/NaPo2006g/s.inc.latex As of: Wednesday 08Feb2006 . Typeset: 7Nov2005 at 22:02 ....
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