from University of Tokyo 1S1 Mathematical Science Foundation: Linear Algebra Sets and Mappings - Sets - In university, the elements of a set are called “元” (pronounced “moto”). - Sets are the same even if they have duplicates or the order is different. - For example, . - This is different from arrays. - When writing , it represents a set consisting of all elements that satisfy the condition P. - It’s like using the .filter() function. - This interpretation is interesting (takker). - Indeed, since , it can also be seen as a filter. - I felt a filter-like taste in writing instead of (blu3mo). - It feels like scanning within the defined universal set. - If the type of x is not determined, we cannot evaluate P(x), so we write it like this (blu3mo). - It’s like static typing in programming (blu3mo). - The main reason is to avoid Russell’s Paradox (takker). - That’s why in the ZFC Axiomatic System, only or are allowed. - (Here, is any logical formula that satisfies ). - I see! I see the connection (blu3mo). - Oh, it’s also fine to write it as . - But it increases the number of characters, so it’s not used much. - I’m curious if it’s not a problem that the set of values for x to check P(x) is not explicitly stated in this notation (blu3mo). - It’s not a problem because it passes the logical formula of the ZFC Axiomatic System (takker). - Like List Comprehension. - represents the range of A and not B. - It’s a new symbol (blu3mo). - There is a dedicated symbol called \setminus (I rewrote it) (takker). - Thank you (blu3mo) (I don’t know anything about TeX). - If you search for it in mathematical notation, it’s listed on Wikipedia (takker). - It’s quite useful (takker). - For example, you can represent “all real numbers except 0” as . - represents the set of all real numbers. - There is a generalization of this set in the direction of the number of variables, called . - is the set of all points on a number line. - is the set of all points on a plane. - Examples include and . - That’s the idea. - If you define it properly, - is an example. - If you have a question, how do you define the parentheses in ? (takker) - It relates to the definition of Cartesian products, as well as mappings, sequences, and vectors. - It’s worth looking into when you have time. - As a general rule, when considering , if you get confused, try to think of it with geometric images using or . - I see (blu3mo). - This kind of tip is helpful, like having a good teacher. - Concrete instantiation of character constants is a very important way of thinking and verification method (takker). - /takker/Concrete instantiation of character constants - Cartesian Product - Define “multiplication” on sets. - . - Even though it’s called multiplication, it’s just combining values. - The use of the exponent notation in is probably because we are dealing with the multiplication symbol here (blu3mo). - That’s right (takker). - It’s nontrivial that . - In the case of , - Since is always false, is always false, and there are no elements. - Ah, I see. It’s true that there is multiplication. I understand the feeling of defining it as a product (blu3mo). - In other words, there is an abstract definition of multiplication, like something x 0/∅/etc = 0/∅/etc (blu3mo). - By the way, the empty set is denoted by (takker). - cf. /takker/Empty Set
from University of Tokyo 1S1 Mathematical Science Foundation: Differential and Integral Calculus