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# b.inc - ² ² p,q ∈ N with q 6 = 0 o For each set below...

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Numbers & Polys MAS3300 Exam-B 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. Write expressions unambiguously e.g, “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 .” B1: Using that N is WOed, prove that there is no negative natnum. ( < Thm2.3 ) B2: YMAssume that N is sealed under addition. Please prove that N is sealed under multiplication. ( < Thm2.7 ) B3: Define a sequence b = ( b 0 , b 1 , b 2 , . . . ) by b 0 := 0 and b 1 := 3 and b n +2 := 7 b n +1 - 10 b n , for n = 0 , 1 , . . . . Use strong induction to prove, for all k [ 0 .. ) , that b k = 5 k - 2 k . B4: Consider these sets: B := - 5 + n n +1 n N ; C := 7 - 1 n n Z + ; D := n · 2 n N ; Q > 0 := n p q p, q N with q 6 = 0 o . For each set below,
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Unformatted text preview: ² ² p,q ∈ N with q 6 = 0 o . For each set below, ± ² ³ ´ circle one of “ WO ” “ Nope ” . In the case of ± ² ³ ´ Nope , provide an explicit inﬁnite de-creasing sequence in the given set. B is WO Nope . C ∪ D is WO Nope . Q > is WO Nope . Zr ( ∞ .. 17 ) is WO Nope . End of Exam-B B1: 100pts B2: 100pts B3: 100pts B4: 60pts Total: 360pts 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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