1. For this question, recall the interval notation for subsets of R: If a, b are real numbers with
a ≤ b then
(a, b) = {x ∈ R : a<x<b} and [a, b] = {x ∈ R : a ≤ x ≤ b}.
Four TAs are discussing the types of sets that they like. If ACR, then
• Niharika likes A if and only if there exists an r > 0 such that for all x ∈ A, (x-r,x+r) ⊆
A.
• Sofia likes A if and only if for every x ∈ A there exists an r > 0 such that (x-r,x+r) ⊆ A.
• Themba likes A if and only if for every x ∈ A there exists an r > 0 such that [x-r, x+r] ⊆
A.
• Yash likes A if and only if there exists an r > 0 and there existsfan x ∈ A such that
[x - r, x + r] ⊆ А.
Be sure to prove the correctness of your answers to the following questions.
(a) Identify which TAs, if any, like the empty set.
(b) Give an example of a set A which every TA dislikes.
(c) Show that Sofia and Themba like exactly the same sets.
(d) Determine whether the following statement is true:
"For every AC R, if Yash likes A then Sofia likes A."
