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9 February 2012Name50 minutesStudent No.62-190-02 — Mathematical Foundations – Test 1— SolutionsWrite only on these pages. There is an extra page at the end, if you need the space.As well as the axioms, you may use any theorem proved in class.1./102./103./104./105./5Page 1 of 7
62– 190–02 Test 1 W20121.(a) LetAbe a subset ofR. Write the symbolic form of the statement“Ahas no largest element”.Soln.¬(∃x∈A,∀y∈A, y≤x).∀x∈A,∃y∈A, y > x.(b) LetAbe the interval [0, π). Prove thatAhas no largest element.Page 2 of 7
62– 190–02 Test 1 W20122.(a) Complete the following definition:Forx, y∈Z,xdividesymeans . . .∃k∈Z, y=kx.(b) Prove, for alln∈N, ifndivides 3, thenn= 1 orn= 3Proof.