I was preparing for my algebra final and I ran into this quotient ring constructed in :

*What even is this?! *

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# A Scary-Looking Quotient Ring …

# Why Is Many-One Reducibility Not A Total Order?

# Integral Domain

# Let’s Partition A Group!

# Shuffle, Shuffle, Shuffle …

# What Does Order Tell You?

# Random Coin

I was preparing for my algebra final and I ran into this quotient ring constructed in :

*What even is this?! *

We use the notation if there’s a many-one reduction from set to set . When I first learned about many-one reduction, the notation made me think imposes a total order among subsets of , i.e. on behaves like on . Unfortunately, only imposes a *partial order* on . In this post, we will discuss an example of two subsets of natural numbers that are not comparable using .

Continue reading “Why Is Many-One Reducibility Not A Total Order?”

We covered Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains before Thanksgiving break. I was slightly stressed during the break because I didn’t grasp the concepts very well. Luckily, when I get stressed, I write notes, and then I’ll be happy again š So here it is:

I think of a group as a bag of items with a rule of smashing two items into one. Sometimes, we don’t want mix everything together in one bag;*Ā *we might want to divide the items into several smaller bags. Let’s talk about two ways of partitioning a group, i.e.Ā dividing the items in the bag.

Groups are like verbs. We can gain more information about a group by looking at how it interacts with other objects.

When I first learned algebra, I thought there’s no reason to believe we can gain structural information of a groupĀ *solely* by looking at the number of elements of the group. Unfortunately (or maybe fortunately :p), I was wrong.

Let’s play a game. I toss a coin twice and ask you, what is the probability of landing two heads? I claim that my coin is a fair coin. Considering I might be lying, what probability should you guess?