Sets and Mappings

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, $\{1,2\}=\{1,2,1\}=\{2,1\}$. - This is different from arrays. - When writing $\{x\in X|P(x)\}$, 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 $\{x\in X|P(x)\}\subseteq X$, it can also be seen as a filter. - I felt a filter-like taste in writing $\{x\in X|P(x)\}$ instead of $\{x|x\in X\wedge P(x)\}$ (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 $\{x\in X|P(x)\}$ or $\{x|\exists y\in Y;Q(x,y)\}$ are allowed. - (Here, $Q(x,y)$ is any logical formula that satisfies $\forall y\in Y\exists !x;Q(x,y)$). - I see! I see the connection (blu3mo). - Oh, it’s also fine to write it as $\{x|x\in X\wedge P(x)\}$. - 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. - $A\setminus B$ 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 $\mathbb{R}\setminus \{0\}$. - $R$ represents the set of all real numbers. - There is a generalization of this set in the direction of the number of variables, called $R^n$. - $R^1$ is the set of all points on a number line. - $R^2$ is the set of all points on a plane. - Examples include $(1,2)$ and $(-\pi ,e)$. - That’s the idea. - If you define it properly, - $R^3:=\{(x,y,z)|x,y,z\in R\}$ is an example. - If you have a question, how do you define the parentheses in $(x,y,z)$? (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 $R^n$, if you get confused, try to think of it with geometric images using $R^1$ or $R^2$. - 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. - $X\times Y:=\{(x,y)|x\in X\wedge y\in Y\}$. - Even though it’s called multiplication, it’s just combining values. - The use of the exponent notation in $R^n$ is probably because we are dealing with the multiplication symbol here (blu3mo). - That’s right (takker). - It’s nontrivial that $A\times \varnothing =\varnothing$. - In the case of $A\times \varnothing :=\{(x,y)|x\in A\wedge y\in \varnothing \}$, - Since $y\in \varnothing$ is always false, $x\in A\wedge y\in \varnothing$ 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 $\varnothing$ (takker). - cf. /takker/Empty Set

from University of Tokyo 1S1 Mathematical Science Foundation: Differential and Integral Calculus