a.inc - T F b “Subtraction is associative.” T F c On R...

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MAS3300 Exam-A Prof. JLF King 4Jun2008 Note. This is an open brain, open ( pristine ) Sigmon- Notes exam. Please write each solution on a separate sheet of paper. Please be sure to write expressions unambigu- ously e.g, the expression “1 /a + b ” should be bracketed ei- ther [1 /a ]+ b or 1 / [ a + b ]. Be careful with negative signs! Every if ” must be matched by a “ then .” A1: Please prove Thm1.4d:(P.2) If e R is an multiplicative-identity then e = 1 . A2: 1.18c:(P.6) Harmonic-Mean Inequality. A3: Prove the triangle ineq., Thm1.20g:(P.6) When x,y R then | x | + | y | > | x + y | . A4: Let * mean “ theorems earlier than (1.15f)”. Using ( * ) prove: Lemma: If z 6 = 0 then z 2 is positive. Now use this and ( * ) to prove that 1 < 0. A5: For each of the following statements in quotes, ± ² ³ ´ circle one of T F . Then provide a proof or a CEX with explicit numbers . a “Addition distributes over mult.”
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Unformatted text preview: T F b “Subtraction is associative.” T F c On R define x / y := ± x · y ² + y . Then “binop / is associative”. T F d On the set of all people, give an example of a bin-rel which is transitive and reflexive , but is not sym-metric . Bonus : Define a binop “ n ” by b n c := 7-h [7 + b ] / [7 + c ] i Prove or give a CEX: “Binop n is associative.” End of Exam-A A1: 45pts A2: 45pts A3: 50pts A4: 50pts A5: 70pts Bonus: 10pts Total: 260pts Print name ............................................. Ord: Honor Code: “I have neither requested nor received help on this exam other than from my professor (or his colleague) .” Signature: ............................................
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