# a < b, b < c implies a < c

A whole day and a night of sleep, and I thought of a strategy: show that A is similar to B; since b < c, if A is such, then the implication follows.

b < c means

1. B is equivalent to a proper subset of C, and
2. C is not equivalent to any subset of B

Since A is equivalent to the proper subset of B, is a subset of B, is essentially B – and C is not equivalent to such subsets of B – then a < c.

# That number called symbol

Algebra, not like in school, starts with sets. These elements associate to elements, set to set, a creative correspondence: chairs to students, objects to positive integers. Counting them, the highest ordinal number becomes the cardinal number, and this is what we see, naked, in every arithmetic operation thence.

The abstraction elevates to the sky: each number represents an abstraction of its class of sets; that is, 3 represents three of a set of anything; we free ourselves from leaks. We define equivalence and relations, addition, subtraction, multiplication and division.

Even the early problems ask for proofs. I’m tempted to jump back into discrete to plow through the proof tips section. But that would start a prerequisite cascade: I would begin with the first sentence of the first chapter, starting over, and exhaust months.

# Reading the well-written

I thought probability theory would be easy, but I was wrong. It feels like black magic, even when you have the solution. My prerequisites were weak sauce, so I knew I had to fix it. I decided to work through “Fundamental Concepts of Algebra.”

Where am I going to find the time to grind through these books? I also need to apply the stuff, so that my programming does not atrophy. I’ve pondered moonlighting as a sys admin at home. Wouldn’t that be a great second job; coming up with learning materials and infrastructure for mom and pop, typical users, and gleaning killer UX experience?

The answer is yes! Of course. I could create a whole ecosystem here. To them, I could be Apple and Google. I could be their rockstar. I could show them my talent. My skills would grow with their computer literacy.

Back to the math: so, you can take any set of objects – logs, men, or rocks – and establish a one-to-one correspondence with the set of positive integers, finite to fit the former’s length, and that cardinal value – that *number* – associates concrete notions with the abstract. Now we can operate on them with all the fine finesse of mathematics.