## 辅导案例-MATH 117

• August 10, 2020

MATH 117 HOMEWORK 1 DUE AUGUST 10TH AT 11:59PM Recommended practice questions (do not hand in). • Exercises 1.2.x, x ∈ {2, 5, 7, 8, 11}. • Exercises 1.3.x, x ∈ {2, 6, 8}. Submit your answers to these questions. You are strongly encouraged to typeset your answers using LaTeX! But if you write them up by hand, make sure your work is neatly and legibly written. (1) Let A, B, C be sets. Give counterexamples to show that (a) and (b) are false, and prove that (c) is true. (a) If A ∩B = A ∩ C then B = C. (b) If A ∪B = A ∪ C then B = C. (c) If A ∩B = A ∩ C and A ∪B = A ∪ C then B = C. [Hint: use small sets for (a) and (b)! Then use the intuition you gain from working out parts (a) and (b) to figure out why (c) must be true.] (2) Let x be a real number. Prove that if x2 and (x+1)3 are both rational numbers, then x is a rational number. (3) Prove by induction that 2n + n2 < 3n, for all n ≥ 2. [Hint: 2n+1 = 2n + 2n.] (4) Consider the set S := {2− 2−n | n ∈ N} ⊆ R. (a) Use the Axiom of Completeness to show that S has a supremum in R. (b) What is sup(S)? Justify your answer. (This means: give a short proof demonstrating that your proposed value of sup(S) really does satisfy the two conditions to be the supremum of S.) (5) Prove that the set T := {x ∈ R | x < 5} ⊆ R has no maximum element.

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