Mathematics · Ch 2 — Relations and Functions

Summary

2.10

An ordered pair $(a,b)$ keeps track of order: $(a,b)=(c,d)$ iff $a=c$ and $b=d$ .
The Cartesian product $A\times B=\{(a,b):a\in A, b\in B\}$ ; $n(A\times B)=n(A)\cdot n(B)$ for finite sets; generally $A\times B\ne B\times A$ .
$\mathbb{R}\times\mathbb{R}$ (written $\mathbb{R}^2$ ) models the coordinate plane; $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ (written $\mathbb{R}^3$ ) models 3-D space.
A relation from $A$ to $B$ is any subset of $A\times B$ , with domain $\subseteq A$ and range $\subseteq$ co-domain $=B$ ; the number of relations from $A$ to $B$ is $2^{n(A)\times n(B)}$ .
A function $f:A\to B$ is a relation in which every element of $A$ has exactly one image in $B$ ; domain $=A$ exactly, range $\subseteq B$ .
Real valued functions studied by formula: find the domain by excluding zero denominators, negative numbers under even roots, and non-positive arguments of logarithms; find the range by reasoning about the formula. …