Set Theory
The Real Numbers
Gaps and Completions
Definition:
Let \(X\) be linearly ordered with respect to \(\preccurlyeq\), and let \(\{A,B\}\subseteq P(X)\) be nonempty subsets of \(X\) such that \(A\cup B=X\), \(A\cap B=\emptyset\), and for every \(a\in A\) and every \(b\in B\), \(a\preccurlyeq b\). Then we say that \((A,B)\) is a cut of \(X\) with respect to \(\preccurlyeq\).
Definition:
Let \(X\) be linearly ordered with respect to \(\preccurlyeq\), and let \((A,B)\) be a cut of \(X\) with respect to \(\preccurlyeq\). If \(A\) has no maximal element, we say that \((A,B)\) is a Dedekind cut. If, in addition, \(B\) has no minimal element, we say that \((A,B)\) is a gap with respect to \(\preccurlyeq\).
Definition:
Let \(X\) be linearly ordered with respect to \(\preccurlyeq\). If \(X\) has no gaps with respect to \(\preccurlyeq\), we say that \(X\) is complete with respect to \(\preccurlyeq\).
Definition:
We denote the set of all Dedekind cuts of \(\mathbb{Q}\) by \(\mathbb{R}\), and call it the set of real numbers.
Definition:
Let \(x\in\mathbb{R}\). Then \(x\) is called a real number, or a real for short.
Definition:
\(\le:=\left\{((A,B),(C,D))\in\mathbb{R}\times\mathbb{R}:A\subseteq C\right\}\).
Corollary:
\(\le\) is an order relation on \(\mathbb{R}\).
Claim:
\(\le\) is a total order relation on \(\mathbb{R}\).
Proof:
Let \(\{(A,B),(C,D)\}\subset P(\mathbb{Q})\times P(\mathbb{Q})\) be Dedekind cuts. We split into cases. If \(A\subseteq C\), we are done. Otherwise there exists \(a\in A\) such that \(a\in\mathbb{Q}\setminus C=D\). Let \(c\in C\). Then \(c<a\), hence \(c\in\mathbb{Q}\setminus B=A\). Therefore \(C\subseteq A\).
Claim:
\(\mathbb{R}\) is complete with respect to \(\le\).
Proof:
Let \((S,T)\in P(\mathbb{R})\times P(\mathbb{R})\) be a Dedekind cut of \(\mathbb{R}\). Write \(A_{S}:=\bigcup\{A:(A,B)\in S\}\), and \(B_{S}:=\mathbb{Q}\setminus A_{S}\). Then \((A_{S},B_{S})\in\mathbb{R}\).
For every \((A,B)\in S\), \(A\subseteq A_{S}\), so \((A,B)\le(A_{S},B_{S})\). Let \((C,D)\in T\). Then for every \((A,B)\in S\), \(A\subset C\), so \(A_{S}=\bigcup\{A:(A,B)\in S\}\subseteq C\). Therefore \((A_{S},B_{S})\le(C,D)\). Thus \((A_{S},B_{S})\) is the least upper bound of \(S\). Since \(S\) has no maximal element, it follows that \(T=\{x:(A_{S},B_{S})\le x\}\), so \((S,T)\) is not a gap.
Definition:
Let \(q\in\mathbb{Q}\). We write \([q]:=(\{p\in\mathbb{Q}:p<q\},\{p\in\mathbb{Q}:q\le p\})\).
Corollaries:
Let \(\{p,q\}\subset\mathbb{Q}\). Then:
- \([q]\in\mathbb{R}\).
- \(p\le q\Longleftrightarrow[p]\le[q]\).
Supremum and Infimum
Definitions:
Let \(A\subseteq\mathbb{R}\) be a nonempty subset, and let \(x\in\mathbb{R}\). Then:
- If for every \(a\in A\), \(x\le a\), we say that \(x\) is a lower bound of \(A\), and that \(A\) is bounded below.
- If for every \(a\in A\), \(a\le x\), we say that \(x\) is an upper bound of \(A\), and that \(A\) is bounded above.
Definitions:
Let \(A\subseteq\mathbb{R}\) be a nonempty subset, and let \(x\in\mathbb{R}\). Then:
- If \(x\) is a lower bound of \(A\), and for every lower bound \(y\) of \(A\), \(x\le y\), we say that \(x\) is the infimum of \(A\), and denote it \(\inf A\).
- If \(x\) is an upper bound of \(A\), and for every upper bound \(y\) of \(A\), \(y\le x\), we say that \(x\) is the supremum of \(A\), and denote it \(\sup A\).
Claim:
Let \(X\subseteq\mathbb{R}\) be nonempty and bounded above. Then \(X\) has a supremum.
Proof:
Write \(A_{X}:=\bigcup\{A:(A,B)\in X\}\), and \(B_{X}:=\mathbb{Q}\setminus A_{X}\). Then \((A_{X},B_{X})\in\mathbb{R}\). For every \((A,B)\in X\), \(A\subseteq A_{X}\), so \((A,B)\le(A_{X},B_{X})\). Let \((C,D)\in\mathbb{R}\) be such that for every \((A,B)\in X\), \(A\subset C\). Then \(A_{X}=\bigcup\{A:(A,B)\in X\}\subseteq C\). Hence \((A_{X},B_{X})\le(C,D)\), so \((A_{X},B_{X})\) is the least upper bound of \(X\).
Definition:
Let \(A\subseteq\mathbb{R}\) be nonempty. If \(A\) is not bounded above, we write \(\sup A=+\infty\). If \(A\) is not bounded below, we write \(\inf A=-\infty\).
Definitions:
Let \(\{x,y\}\subset\mathbb{R}\). Then:
- \(x+y:=\sup\left\{[p+q]:p\in\mathbb{Q},q\in\mathbb{Q},[p]\le x,[q]\le y\right\}\).
- \(x\cdot y:=\sup\left\{[p\cdot q]:p\in\mathbb{Q},q\in\mathbb{Q},[p]\le x,[q]\le y\right\}\).
- \(-x:=\sup\{[q]:q\in\mathbb{Q},[q]+x\le[0]\}\).
- \(|x|:=\begin{cases}-x & x\le[0]\\ x & \text{otherwise}\end{cases}\)
- If \(x\neq0\), \(x^{-1}:=\sup\{[q]:q\in\mathbb{Q},[q]\cdot x\le[1]\}\).
- If \(y\neq0\), \(\frac{x}{y}:=x\cdot y^{-1}\).
- \(\sqrt{x}:=\sup\left\{[q]:[q\cdot q]\le x\right\}\).
Corollaries:
Let \(\{x,y,z\}\subset\mathbb{R}\). Then:
- \(x+y=y+x\).
- \((x+y)+z=x+(y+z)\), and we may write \(x+y+z\) briefly.
- \(x+0=x\).
- \(x-x=0\).
- \(x\cdot y=y\cdot x\).
- \((x\cdot y)\cdot z=x\cdot(y\cdot z)\), and we may write \(x\cdot y\cdot z\) briefly.
- \(x\cdot0=0\).
- \(x\cdot1=x\).
- \(x\cdot(-1)=-x\).
- \((-x)\cdot(-y)=x\cdot y\).
- \((-x)\cdot y=x\cdot(-y)=-(x\cdot y)\).
- \(x\cdot(y+z)=x\cdot y+x\cdot z\).
- \(x<y\Longleftrightarrow x+z<y+z\).
- If \(z>0\), then \(x<y\Longleftrightarrow x\cdot z<y\cdot z\).
- If \(z<0\), then \(x<y\Longleftrightarrow y\cdot z<x\cdot z\).
Notation:
Let \(q\in\mathbb{Q}\) be rational. From now on we will no longer distinguish between \(q\) and \([q]\). For all practical purposes, we assume \(\mathbb{Q}\subset\mathbb{R}\).
Definitions:
Let \(a<b\) be real numbers. Then:
- \([a..b]:=\{x\in\mathbb{R}:a\le x\le b\}\)
- \((a..b]:=\{x\in\mathbb{R}:a<x\le b\}\)
- \([a..b):=\{x\in\mathbb{R}:a\le x<b\}\)
- \((a..b):=\{x\in\mathbb{R}:a<x<b\}\)
- \((-\infty..b]:=\{x\in\mathbb{R}:x\le b\}\)
- \((-\infty..b):=\{x\in\mathbb{R}:x<b\}\)
- \([a..+\infty):=\{x\in\mathbb{R}:a\le x\}\)
- \((a..+\infty):=\{x\in\mathbb{R}:a<x\}\)