(1) Write the Cauchy theorem (without proof).

1. Let DC R. Write down precise mathematical formulations of the following statements, using quantifiers (V, 3): (a) D is bounded above. (b) D has a maximum element. (c) D does not have a maximum element.

I don't understand this question at all.  I've asked it a few times, and have gotten what look like a correct solutions, but I don't understand the reasoning behind many of the steps.  I'm asking it again, hoping to get an even more thorough explanation.  I'm sorry, it just isn't making sense to me.  Thanks for your time.

Label the following statement as True (T) or False (F): If L ∈ NP then L is decidable.

a 3 non ata gio max.: ntrassegna da ecedente The inequality √x-1 √x-2 Successivo > 0 is satisfied for x in O (A) [0, 1) O (B) [0, 1) n (4, +∞0) O (C) (4, +∞0) O (D) [0, 1] U [4, +∞0) O (E) [0, 1) U (4, +∞) 31097

Let x = (x1, x2, x3) ER³ and suppose you are given a subset UCR³ defined by U = {ER³ : x₂ < 5} Which of the following statements are necessarily true? (i) The complement of U contains a boundary point. (ii) The complement of U contains no interior points. (iii) U is not an open set. (iv) U is not a closed set. (A) (iii) only (B) (i) and (iv) (C) (ii) only (D) (i), (iii), and (iv) (E) (ii) and (iii) (F) (iv) only (G) (i) only (H) (iii) and (iv)

8. Find the reflexive, symmetric and transitive closure of R if MR is 1 0 (1 0 1 0 1 1 0 0 1 0 0/

(a) Prove that (-∞, 0) U (1, 2) U (2, ∞) is an open set.

4. (a) Use the Method of Smallest Counterexamples based on the Well-Ordering Prin- ciple to prove the following: For all n E N, n! ≤n". (b) Let a and b be nonzero integers. Use the Well-Ordering Principle to show that there is a smallest positive linear combination of a and b.

Nine Starbucks are located in a 3 mile by 3 mile square area in which there is a 1 mile by 1 mile square lake in the at the center of one of the sides of the area, as in the diagram below. There are no Starbuck's in the lake! (a) Prove that there is at least a pair of Starbuck's that are closer to each other than 1.5 miles. (b) Demonstrate that, on the other hand, it is possible to locate eight Starbucks in the square area (minus the lake) that are greater than or equal to 1.5 miles from each other. Explain your reasons for your answers in detail. 1 mile by 1 mile lake 3 miles 3 miles 1 mile 1 mile