Here is a brief overview of basic results and definitions concerning sets and the reals that should be assumed for this course.
DefinitionA.1.
\(\mb R\) is the set of real numbers.
\(\mb Z\) is the set of integers.
\(\omega=\setBuilder{z\in\mb Z}{z\geq 0}=\setList{0,1,2,\dots}\) is the set of natural numbers, which includes zero.
\(\mb Q=\setBuilder{\frac{z}{n+1}}{z\in\mb Z,n\in\omega}\) is the set of rational numbers.
TheoremA.2.
De Morgan's Laws: Let \(\mc A\) be a collection of subsets of \(X\text{.}\)
\begin{equation*}
X\setminus\bigcup_{A\in\mc A}A=\bigcap_{A\in\mc A}(X\setminus A)
\end{equation*}
\begin{equation*}
X\setminus\bigcap_{A\in\mc A}A=\bigcup_{A\in\mc A}(X\setminus A)
\end{equation*}
TheoremA.3.
The Archemedian Property of the real numbers guarantees that for each positive real number \(x>0\text{,}\) there exists a natural number \(n\in\omega\) such that \(\frac{1}{n}\lt x\text{.}\)
TheoremA.4.
Let \(S\subseteq \mb R\) be a set of real numbers with a lower bound. Then there exists a greatest lower bound (a.k.a. infimum) \(\glb S=\inf S\text{.}\)
Let \(S\subseteq \mb R\) be a set of real numbers with a lower bound. Then there exists a least upper bound (a.k.a. supremum) \(\lub S=\sup S\text{.}\)
DefinitionA.5.
The cardinality of a finite set is the natural number that is equal to the number of elements it contains. For example, \(S=\setL{-1,\setL{5,-e},\pi}\) has cardinality \(|S|=3\text{.}\)
A set \(S\) that has an injection with \(\omega\) is said to be countable and have cardinality \(|S|\leq\aleph_0\text{.}\) If a bijection exists, the set is countably infinite and has cardinality \(|S|=\alpha_0\text{.}\)
A set \(S\) that has no injection with \(\omega\) is said to be uncountable and have cardinality \(|S|>\aleph_0\text{.}\)
A set \(S\) that has a bijection with \(\mb R\) is said to be continuum-sized and have cardinality \(|S|=\mathfrak c\text{.}\)
LemmaA.6.
For each \(a,b\in\mathbb R\cup\setL{-\infty,\infty}\text{,}\) the set \((a,b)\cap\mathbb Q\) is infinite and countable, and the set \((a,b)\setminus\mathbb Q\) is infinite and uncoutnable. In particular, every subinterval \((a,b)\) of \(\mathbb R\) contains a rational from \(\mathbb Q\) and an irrational from \(\mathbb R\setminus\mathbb Q\text{.}\)
LemmaA.7.
For any functions \(f:A\to B,g:B\to C\) and \(Z\subseteq C\text{,}\)\((g\circ f)^\leftarrow[Z]=f^\leftarrow[g^\leftarrow[Z]]\text{.}\)
