# Gracious Living

Notation

In order to make my posts more readable, I keep re-defining notation when we’ve spent a while away from it.  To save time, here are some pretty common notations that I haven’t really taken the time to define.

General/Set Theory

$\forall,\exists$ — “for all” and “there exists a” respectively.

$\mathbb{N,Z,Q,R,C}$ — the sets of natural numbers (0,1,2,…), integers (…,-1,0,1,…), rationals (fractions), reals (uh… rationals and irrationals, the whole number line), and complex numbers ($a+bi$ where $a,b$ are real, $i^2=-1$) respectively.

$\mathbb{H}$ — the set of quaternions.  A bit more complicated to describe, it’s a noncommutative “division ring” (almost a field) of things of the form $a+bi+cj+dk$, where $a,b,c,d\in\mathbb{R}$ and multiplication is defined distributively by the “$Q_8$” entry below.

$\sum$ — the sum of a series of numbers.

$\times,\prod$ — either the product of numbers or the Cartesian product of sets

$\cup,\cap,-$ — respectively the union, intersection, and difference of sets (last one is also a minus sign).  If you’re shaky on this, I suggest you read the first few posts.

$\sqcup,\bigsqcup$ — the disjoint union of sets.  This means we take the union but force elements from different sets to look different.  That is, $\{1,2,3\}\sqcup\{1,2\}\cong\{1,2,3,1^\prime,2^\prime\}$.  One standard way of doing this is to have, say, $A\sqcup B=A\times\{0\}\cup B\times\{1\}\subset (A\cup B)\times\{0,1\}$, and so on — so the above example would become $\{(1,0),(2,0),(3,0),(1,1),(2,1)\}$.

$\in$ — “is an element of.”

$\subset$ — “is a subset of.”  I allow non-proper subsets here, and use $\subsetneq$ for proper subsets.

$\mathbb{R}^n$ — the set of ordered $n$-tuples of reals, like $(a_1,a_2,\dotsc,a_n)$.  That is, we’re taking the Cartesian product of $\mathbb{R}$ with itself $n$ times.  If you’re geometrically minded, $\mathbb{R}^2$ is the plane and $\mathbb{R}^3$ is space.

intervals — subsets of $\mathbb{R}$ between two points.  A parenthesis at one end means we don’t include that endpoint, a bracket means we do.  So $(0,1]=\{x\in\mathbb{R}:0.

$!$ — the factorial function, defined by $n!=n(n-1)\dotsm\cdot 2\cdot 1$.

$\cong$ — “is isomorphic to,” so pretty much “the same as” under whatever level of detail we’re talking about.  For sets without any other structure, this just means a bijection exists, hopefully natural.

$\mathcal{P}(A)$ — the power set, or set of subsets, of $A$.

$A^B$ — the set of functions from $B$ to $A$.  In particular, if we treat $n$ as the set $\{0,\dotsc,n-1\}$, then $A^n$ is in bijection with the $n$-fold Cartesian product of $A$ with itself, and $2^A$ is in bijection with the power set of $A$.

$\circ$ — composition of functions.  $(g\circ f)(x)=f(g(x))$.

$|A|$ — the cardinality of $A$, its equivalence class under bijection.

$\aleph_0$ — the smallest infinite cardinality, that of the natural numbers.

$\aleph_\alpha$ — the class of cardinalities is well-ordered, so this is just the $\alpha$th one after $\aleph_0$.  $\alpha$ can be any ordinal.

$c$ — the cardinality of the reals.  Stands for “continuum.”

$\beth_\alpha$ — take $\beth_0=\aleph_0, \beth_{S\alpha}=2^{\beth_\alpha},\beth_{\lim\alpha}=\lim\beth_\alpha$.  So in particular, $\beth_1=c$.  Again, $\alpha$ can be any ordinal, by transfinite induction.  Under the continuum hypothesis, $\beth_1=\aleph_1$, and under the generalized version, $\beth_\alpha=\aleph_\alpha$ for all $\alpha$.

$\omega$ — the first infinite ordinal, the order-type of the natural numbers.

$\epsilon_0$ — the first ordinal that can’t be produced from $\omega$ using primitive recursive functions.

$\omega_1^{CK}$ — the first ordinal that can’t be produced from $\omega$ using definable functions.  Still countable, though.

$\omega_\alpha$ — the smallest ordinal of cardinality $\aleph_\alpha$.

addition, multiplication, exponentiation — mean the corresponding operation on ordinals or cardinals, depending on what we’re applying them to.  Figure it out, people!

Topology

$I$ — the closed interval $[0,1]$.

$X^n$ — in general, meaning the product topology.  So $I^2=\{(x,y)\in\mathbb{R}^2:0\le x,y\le 1\}=$ the closed unit square as a subspace of $\mathbb{R}^2$.

$S^n,D^n,B^n,M^n$ — exceptions to the above.  If $M$ is a manifold, writing it as $M^n$ means it has dimension $n$.  In particular, $S^n$ is the $n$-sphere $\{(x_1,\dotsc,x_{n+1})\in\mathbb{R}^{n+1}:\sqrt{x_1^2+\dotsb+x_{n+1}^2}=1\}$.  So $S^0$ is the two points $\{-1,1\}$; $S^1$ is the unit circle; $S^2$ is the unit sphere; and so on.  Likewise, $D^n$ or $B^n$ is the disk/ball enclosed by $S^{n-1}$: it’s $\{(x_1,\dotsc,x_n)\in\mathbb{R}^n:\sqrt{x_1^2+\dotsb+x_n^2}\le 1\}$.  Notice this always includes its boundary.  We want, say, the 2-sphere to enclose the 3-ball because the 3-ball is actually 3-dimensional and the 2-sphere is only 2-dimensional, so the numbers are a bit confusing.  In general topology, we won’t really care about the exact size or even shape of the sphere or ball since everything’s up to homeomorphism; but when we do care, you’re to assume that they are of unit size and “standard.”  For example, in differential geometry, the sphere is assumed to have the standard metric and constant curvature $1$.

$\approx$ — “is homeomorphic to.”

$\simeq$ — for two functions, “is homotopic to;” for two spaces, “is homotopy-equivalent to.”

Algebra

$\le$ — “is a subgroup of” for groups, “is an ideal of” for rings, and so on.

$\unlhd$ — “is a normal subgroup of.”

$\cong$ — “is isomorphic to.”

$\times,*,\rtimes$ — respectively, direct, free, and semidirect products.  For the semidirect product, the normal subgroup is the one on the left, so the triangle in the symbol looks like a “normal subgroup” symbol.

$S_n$ — the symmetric group on $n$ symbols, that is, the group of permutations of $n$ distinct objects, of order $n!$.

$A_n$ — the alternating group on $n$ symbols, that is, the group of even permutations of $n$ distinct objects, of order $n!/2$.

$D_n$ — the dihedral group, the group of symmetries of a regular $n$-gon.

$Q_8$ — the “quaternion group,” or group of quaternionic units.  This is $\{1,i,j,k,-1,-i,-j,-k\}$ with the relations $i^2=j^2=k^2=ijk=-1$.  $1,i,j,k$ are a basis for $\mathbb{H}$.

$V_4$ — the Klein four group, $\mathbb{Z}_2\times\mathbb{Z}_2$.

Analysis

$C^k(\mathbb{R}^n)$ — the ring of $n$ times continuously differentiable functions on $\mathbb{R}^n$.

$*$ — convolution of functions on $\mathbb{R}^n$, defined by $(f*g)(x)=\int_{\mathbb{R}^n}f(y)g(x-y)\,dy$.  Note this isn’t guaranteed to exist without certain decay or integrability conditions on $f$ and $g$.

$\int d\mu$ — integration with respect to a measure.