Question

Please answer Problems 1 and 2 thoroughly.

Problem 1: Let X be a set. Define a partial ordering ≤ on P(X) by A ≤ B if and only if A ⊆ B. We stated the following two facts in class. In this exercise you are asked to give a formal proof of each:

(a) (1 point) If A, B ∈ P(X), then sup{A, B} exists, and sup{A, B} = A ∪ B.

(b) (1 point) If A, B ∈ P(X), then inf{A, B} exists, and inf{A, B} = A ∩ B.

Problem 2: For each of the following subsets of R, find the sup in the universe R or state that it does not exist. No proofs or explanations necessary; just provide the answers.

(a) (0.5 points) S1 = [3, 4)

(b) (0.5 points) S2 = (2, ∞)

(c) (0.5 points) S3 = ?3n−2 : n ∈ N? 5n

(d) (0.5 points) S4 = [2, 9]

Answer 1