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Transcript

1. Prove the second part of De Morgan’s Laws, namely for sets A and B A ∪ B = A ∩ B. 2. Show that one of the digits 1, 2, . . . , n occurs infinitely often in the decimal expansion of π 3. Decide whether the following sets are bounded by trying to find a upper/lower bound (note: it does not need to be the least upper bound). ) ( n X 1 n∈N (a) k2 k=1 ( (b) ) n X 1 n∈N k k=1 (c) 1 x sin x x ∈ (0, 2π) (d) {x cos (x) | x ∈ [−π, π]} 4. Show that the set of positive real numbers has no smallest element. 5. Find the least upper bound for the following sets (a) {(−1, 3) ∪ [2, π)} (b) y | y = x2 + 2x + 1 | x ∈ (−2, 2) (c) (d) (−1)n 1 n ∈ N n n o 2 e−x | x ∈ R 6. Suppose S, T ⊂ R are bounded subsets so that Ls ≤ x ≤ US for all x ∈ S and LT ≤ y ≤ UT for all y ∈ T . What can you say about bounds for S ∪ T and S ∩ T . 7. Prove that if n is an even integer, then n2 + 2n + 1 is an even integer.