u/Aggressive-Food-1952

I’m reading an intro text on real analysis and I stumbled across a connection. Say I have a set X with a multiplicative left coset and additive right coset, aX + b.

Then sup(aX + b) = asupX + b, right? This is the same for expected value of a random variable. E[aX + b] = aE[x] + b.

Does this have any meaning or is it just a coincidence? Can I even define such a coset aX + b? (Am I even allowed to call it a “coset”?)

I have a very long, boring summer ahead of me with no plans whatsoever. So I’m thinking of doing some light reading, and I was wondering if you guys had any recommendations for some books.

I’ve read Hammack’s Book of Proof, which I enjoyed. It was a very beginner-friendly book in my opinion, so I’m looking for something similar.

I’m thinking of something maybe analysis-related since I am taking that course next semester. (And hopefully something I can either pick up at the library, purchase for a cheap price, or find online).

Any tips?

Toyota corolla 2021 SE. Rock hit the windshield and cracked it pretty badly. The crack is now about 2 feet from the corner and another started to form. Is aftermarket good for cracked windshields? Is it worth waiting and paying for OEM?

Just finished finals week. It went great academically, but mentally it drained me. However, I am also very sad to be not in school for the summer. I love my college town and my friends, and staying busy helps me deal with my anxious thoughts. I feel really bummed to be going back home. I also just had something happen to my car, so that’s another financial stressor and adds to my anxiety. I also don’t have anything to really look forward to this summer.

Does anyone have tips on how to get through this? My brain is telling me that this goodbye to my friends is permanent, even though it’s not and I’ll see them in 3 months. It’s just that during the summer, I have so much free time that I’m constantly overthinking everything and being anxious at the tiniest things.

In abstract algebra we learned that, say, if two objects are isomorphic, they are essentially the same and all (I think) properties are translated. But when is this not true? Are there general properties that are not preserved or does the definition of an isomorphism automatically force it to hold?

I ask because we learned that V^n is isom. to R^n (which I think is so cool that you can just study the plane as vectors and properties of spaces).

Following this, do the normal notions of vector spaces hold in R^n? Like V^3 and R^3, clearly the dimension is the dimension, but what do the other things correspond to? Like a linear transformation or a basis or subspaces or dot products, so many things!!

I don’t understand it. The basic definitions don’t satisfy me—they seem like a massive oversimplification (but I could be wrong and overthinking it).

Can someone help me understand it? I have strong backgrounds in calculus, linear algebra, statistics and probability, and theoretical mathematics, if that helps.

I recently stumbled across the fact that -i = 1/i… is this true for any other numbers?

If such an equality exists, wouldn’t that cause problems? I’ve never taken a complex analysis class, but it’s such a unique identity that it has me questioning what its uses could be.

A little embarrassing to admit, but I honestly never understood induction. Sure I can apply it, but it just seems illogical to me. I’m wondering if anyone can clear it up for me.

Let S(n) be the inductive hypothesis. We assume it’s true to begin with, then we try to show that S(n) implies S(n+1). But this doesn’t make sense to me. Are we not assuming that the statement is valid to begin with if we assume S(n)? Isn’t it circular reasoning? I’ve heard induction has something to do with the definition of the natural numbers and the successor function. What is this about?

In a field, all nonzero elements have a multiplicative inverse... but why? Why doesn’t 0 have an inverse?

Sure, we can’t divide by 0, but in an abstract sense, we invented 0 as an element of the field. In any arbitrary field, “0” as a number might not even exist since we don’t know what these elements are!

So without saying “dividing by 0 is illegal,” is there a formal reason why we don’t have an inverse for 0? Is it just due to convention?