I'm learning abstract algebra and we just defined rings. I understand the formal definition: A ring (R, +, ·) is a set with two binary operations such that: 1. (R, +) is an abelian group 2. (R, ·) is associative 3. Distributive laws hold But why do we study rings? What are the most important examples? And what is the difference between a ring, a domain, and a field? Also, what is ℤₙ (integers modulo n) as a ring?

I am studying real analysis and I'm learning about the Baire Category Theorem. It states: In a complete metric space, the intersection of countably many dense open sets is dense. Equivalently: A complete metric space cannot be expressed as a countable union of nowhere dense sets. My questions: 1. What does "nowhere dense" mean? How is it different from "not dense"? 2. What is the intuition behind the Baire Category Theorem? 3. What are some stunning applications of the theorem? 4. How do I prove that ℚ is not a G_δ set using Baire? 5. How does Baire imply that there exist continuous functions that are nowhere differentiable? I want to understand why this theorem is so fundamental.

I'm studying linear algebra and I can find the inverse of a 2 \ imes 2 matrix easily using the formula A^(-1) = 1/det(A) d & -b \ -c & a. But for 3 \ imes 3 matrices, the formula involving cofactors and the adjugate is very tedious. I've heard that Gauss-Jordan elimination (row reduction) is more efficient. Could someone show the complete row reduction process to find the inverse of: A = 1 & 2 & 3 \ 0 & 1 & 4 \ 5 & 6 & 0

In my Real Analysis class, we proved that ℚ is countable by constructing a bijection with ℕ. The standard proof uses the diagonal array argument: 1/1 & 1/2 & 1/3 & 1/4 & ⋯ 2/1 & 2/2 & 2/3 & 2/4 & ⋯ 3/1 & 3/2 & 3/3 & 3/4 & ⋯ 4/1 & 4/2 & 4/3 & 4/4 & ⋯ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ Then we traverse along diagonals and skip duplicates. But I'm confused: doesn't Cantor's diagonal argument also show that ℝ is uncountable? Why does the same diagonal argument work for countability of ℚ but uncountability of ℝ?