Sets and Maps: Ordinary Notation

¶ This setting extends Class Theory NBG: the Axioms, Standing Conventions and Basic Notation: its axioms, conventions and notation are in force, as modified below.

¶ Letters of either case may denote sets, while a class that is not a set is still denoted by an upper-case letter; this relaxes Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §notation.

¶ Classes are formed only by class abstraction from predicative formulas, as in Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §comprehension. For a set $A$ and a property $P$ of its elements so expressed, $\{x\in A:P(x)\}$ is as in Class Abstraction: the Class of All Sets Satisfying a Predicative Formula §restricted; it is a set by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass.

¶ A subset of a set is as in Subclasses and Subsets §subclass, and nonempty abbreviates other than $\emptyset$. For sets $A$ and $B$, $A\cup B$, $A\cap B$ and $A\setminus B$ are sets by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §binary-union and Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass. $A\times B$ is the Cartesian product, a set with the elements described in Membership in a Cartesian Product, and the Cartesian Product of Two Sets Is a Set §membership and Membership in a Cartesian Product, and the Cartesian Product of Two Sets Is a Set §set; for an ordered pair $p$, $p_{1}$ and $p_{2}$ are its components.

¶ Let $A$, $B$ and $C$ be sets. A map $f:A\to B$ and its value $f(x)$ at $x\in A$ are as in Functions, Values of a Function, and Functions from One Class to Another §map and Functions, Values of a Function, and Functions from One Class to Another §value; such a map is a set by Images of Sets under Functions Are Sets, and a Function with a Set Domain Is a Set §function-set. For a map $g:A\times B\to C$, $g(x,y)$ stands for $g((x,y))$. When an expression $t(x)$, built from defined set symbols used properly, lies in $B$ for every $x\in A$, the map $x\mapsto t(x)$ from $A$ to $B$ is the unique map with value $t(x)$ at each $x$, given by Maps and Relations Given by Formulas §map; when $t(x,y)\in C$ for all $x\in A$ and $y\in B$, the map $(x,y)\mapsto t(x,y)$ from $A\times B$ to $C$ is given likewise by Maps and Relations Given by Formulas §binary. $B^{A}$ is the set of maps from $A$ to $B$, as in The Class of All Functions from One Set to Another §functions and The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set §set.

¶ A map on a set $I$ written with subscripts, $i\mapsto a_{i}$, is also called a family $(a_{i})_{i\in I}$.

¶ For a map $f:A\to B$, a subset $S$ of $A$ and a subset $T$ of $B$, the image $f(S)$ and the preimage $f^{-1}(T)$ are $f[S]$ and $f^{-1}[T]$ of The Image and the Preimage of a Class under a Class §image and The Image and the Preimage of a Class under a Class §preimage. The image is a set by Images of Sets under Functions Are Sets, and a Function with a Set Domain Is a Set §image-set; the preimage is a subclass of $A=\operatorname{dom}f$ by The Image and the Preimage of a Class under a Class §preimage and Relations, Domain, Range, Inverse and Composition §domain, hence a set by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass. If a set $S$ is both an element and a subset of $A$, then $f(S)$ denotes the value of $f$ at $S$ and the image is written $f[S]$; likewise for $f^{-1}$ when $f$ is a bijection. Images of unions, of subsets and under composites obey Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §union, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §monotone and Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §composition, and for injective $f$ also Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §membership, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §intersection, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §difference and Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §inclusion.

¶ For a map $f:A\to B$ and a subset $S$ of $A$, the restriction of $f$ to $S$ is written $f|_{S}$. The composite $g\circ f$ of $f$ with a map $g:B\to C$ and the identity $\mathrm{id}_{A}$ are as in Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction §composition and The Identity on a Class §identity. Injective, surjective and bijective maps are as in Injective, Surjective and Bijective Functions between Classes, and the inverse of a bijection is that of Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction §inverse. Maps obey the rules of Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction.

¶ Binary operations $\ast$ on a set and the notation $x\ast y$ are as in Binary Operations on a Set §notation.

¶ Partial and total orders $\le$ on a set, and the strict relation $<$, are as in Partial and Total Orders on a Set and the Associated Strict Relation §partial, Partial and Total Orders on a Set and the Associated Strict Relation §total and Partial and Total Orders on a Set and the Associated Strict Relation §strict, and obey the rules of Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders; for $x$ and $y$ in the set, $x\ge y$ means $y\le x$, and $x>y$ means $y<x$. For a set $S$ introduced as a subset of a set $X$ with a partial order, upper and lower bounds, boundedness, least and greatest elements, $\min S$, $\max S$, $\sup S$ and $\inf S$ are as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order, formed in $X$; well-orders are as in Well-Orders on a Set §well-order.