I'm studying abstract algebra and I understand the group axioms: 1. Closure: a · b ∈ G for all a, b ∈ G 2. Associativity: (a · b) · c = a · (b · c) 3. Identity: There exists e ∈ G such that e · a = a · e = a 4. Inverses: For each a, there exists a^(-1) such that a · a^(-1) = a^(-1) · a = e But I've heard that sometimes axioms 3 and 4 can be weakened. Specifically, if we have a semigroup (closure + associativity) with a left identity and left inverses, that is sufficient for a group. Could someone prove this? Also, what is the difference between a group, a monoid, and a semigroup?

I'm studying calculus and I'm learning about volumes of solids of revolution. I know there are two methods: Disk method: V = π ∫_a^b [R(x)]² dx when rotating around the x-axis Washer method: V = π ∫_a^b [R(x)² - r(x)²] dx when there is a hole But I'm confused about when to use each method. Also, when should I integrate with respect to x versus y? Could someone explain with the example of finding the volume of the solid obtained by rotating the region bounded by y = √x, y = 0, and x = 4 about the x-axis?

I am studying complex analysis and I need to understand the Cauchy-Goursat theorem. The theorem states: If f(z) is analytic in a simply connected domain D, then for any closed contour C in D: ∮_C f(z) dz = 0 My questions: 1. What does "simply connected" mean exactly? What's the difference between simply and multiply connected domains? 2. If f has a singularity inside C, what happens? Can I still use a modified version of the theorem? 3. How do I evaluate ∮_C 1/z dz where C is the unit circle? The function has a singularity at z = 0, so the theorem doesn't apply directly. 4. What is the deformation of contours principle? I need intuitive explanations with examples.

The Central Limit Theorem (CLT) is often described as the most important theorem in statistics. I know the basic statement: For i.i.d. random variables X₁, X₂, \ldots, Xₙ with mean μ and variance σ² < ∞: \frac \barXₙ - μ σ/√n \xrightarrowd N(0, 1) as n \ o ∞. But why is this so remarkable? And how large does n need to be for the approximation to be good? Does it work for any distribution?

I'm studying number theory and I need to solve linear congruences like: 17x ≡ 3 (mod 29) I know I need to find the modular inverse of 17 modulo 29 using the Extended Euclidean Algorithm. But I get confused with the back-substitution step. Could someone show me how to solve this specific congruence step by step, explaining how the Extended Euclidean Algorithm works?

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