assignments

Assignment 1

• Prove that \(2 \cdot_{\mathbb{R}} 3 = 6\).
• Build the complex numbers out of the reals in a similar spirit to the constructions we've seen. Prove some fraction of properties you want them to have and discuss at least a little why you expect that your construction has the others too.
• Exercise 2.6.
• Write up an exercise of your choice that we've talked about in class from Chapters 2 and 3.

Assignment 2

• 5.28 from Goldrei.
• Let \(A\) and \(B\) be sets. What is \( \cap \cap \langle A,B \rangle \)?
• Show that \(A\) is a subset of \(B\) if and only if the power set of \(A\) is a subset of the power set of \(B\).
• Show that AC is equivalent to the statement "If R is an equivalence relation on X then there is a set containing exactly one representative of each equivalence class of R".
• Show that there cannot be three sets \(A\), \(B\) and \(C\) with \( A \in B \), \(B \in C \) and \(C \in A\).
• Write up an exercise of your choice that we've talked about in class from Chapters 4 and 5.

Assignment 3

• 6.40, 6.59, 8.12, 9.17 from Goldrei
• Write up one or more exercises of your choice from Chapters 6-9.
• Prove that every vector space has a basis using transfinite induction (and not using Zorn's Lemma). Link for other possible questions.

Final Assignment

• From Goldrei: 4.53, 5.16, 8.10 (Note: I don't /think/ that either of you chose one of these for a write-up earlier. If you did, say so, and substitute it for another one on a similar topic).
• From Sets, Logic and Categories: 5.2, 5.3, 5.4.