Today's soundtrack is Marc Papeghin: When Dream and Horn Unite, a horn-based cover album of many different Dream Theater songs and motifs. It's a really cool little album (under 15 minutes in length); it neither underplays nor embellishes the original works: I think that they've done very well with the arrangements.
This evening, I'll be going through Chapter 20 of Basic Math & Pre-Algebra For Dummies, "Setting Things Up With Basic Set Theory."
I found all of my set theory symbols here.
The chapter begins by saying that "set theory is the foundation for everything in math" (p. 271) - no small claim! I'm intrigued already.
Sets are collections "of things, in any order" (p. 272). There are three main ways to list sets: 1) put a list of items inside of {} braces, 2) a verbal description, and "[w]riting a mathematical rule (set-builder notation)" (p. 272). Three examples of these given in the book are as follows: "A = {Empire State Building, Eiffel Tower, Roman Colosseum} [...] B = {Albert Einstein's intelligence, Marilyn Monroe's talent, Joe DiMaggio's athletic ability, Sen. Joseph McCarthy's ruthlessness} [...] C = the four seasons of the year" (p. 272).
The items inside a set are called elements, or members. We use the symbol "∈" to represent things that are elements of a set, and "∉" represents elements that are not a part of the set.
The "cardinality" of a set is simply how many elements are inside of it. If a set has five elements, its cardinality is five.
If two or more sets have the same elements, they are equal, even if there are more instances of the same element in one than the other.
If a small set's elements can all be found inside of a larger set, that small set is a subset of the larger set, represented by the symbol "⊂"; interestingly, "every set is a subset of itself" (p. 274).
An empty set, or null set, has no elements; we use "Ø" to represent an empty set. Because an empty set has nothing inside of it, every empty set is technically a subset of every other set.
There are four operations that can be performed on sets: "union, intersection, relative complement, and complement" (p. 275).
To find the union (represented by "∪") of two sets, we simply combine their elements, taking out any doubles.
To find the intersection (represented by "∩") of sets, we list the elements that they have in common.
To find the relative complement (represented by "-") of two sets, we start in the first set and take out each element that also exists in the second set.
To find the complement of a set (represented by a single quotation mark ' ), we need to list everything that is not in the set. To do so, we first need to define the universal set (represented by "U"), which is every possible element. So if I made a set like this: C = {Alberta, Quebec, Saskatchewan}, then found its compliment, I would first define the universal set thus: U = {Alberta, British Columbia, Manitoba, New Brunswick, Newfoundland and Labrador, Nova Scotia, Ontario, Prince Edwared Island, Quebec, Saskatchewan}; next, I would identify the complement of C like this: C' = {British Columbia, Manitoba, New Brunswick, Newfoundland and Labrador, Nova Scotia, Ontario, Prince Edward Island}.
