∃, ∃!, ∀ mathematic

[ref: https://en.wikipedia.org/wiki/List_of_mathematical_symbols]

Symbol
in HTML
Symbol
in TeX
NameExplanationExamples

{\displaystyle \exists \!\,}\exists \!\,
there exists;
there is;
there are
∃ xP(x) means there is at least one x such that P(x) is true.∃ n ∈ ℕ: n is even.
∃!
{\displaystyle \exists !\!\,}\exists !\!\,
there exists exactly one
∃! xP(x) means there is exactly one x such that P(x) is true.∃! n ∈ ℕ: n + 5 = 2n.

{\displaystyle \forall \!\,}\forall \!\,
for all;
for any;
for each;
for every
∀ xP(x) means P(x) is true for all x.∀ n ∈ ℕ: n2 ≥ n.



{\displaystyle \in \!\,}\in \!\,

{\displaystyle \notin \!\,}\notin \!\,
is an element of;
is not an element of
everywhere, set theory
a ∈ S means a is an element of the set S;[7] a ∉ S means a is not an element of S.[7](1/2)−1 ∈ ℕ

2−1 ∉ ℕ
{\displaystyle \not \ni }\not \ni
does not contain as an element
S ∌ e means the same thing as e ∉ S, where S is a set and e is not an element of S.
{\displaystyle \ni }\ni
such that symbol
such that
often abbreviated as "s.t."; : and | are also used to abbreviate "such that". The use of ∋ goes back to early mathematical logic and its usage in this sense is declining.Choose {\displaystyle x}x ∋ 2|{\displaystyle x}x and 3|{\displaystyle x}x. (Here | is used in the sense of "divides".)
contains as an element
S ∋ e means the same thing as e ∈ S, where S is a set and e is an element of S.




N
{\displaystyle \mathbb {N} \!\,}\mathbb {N} \!\,

{\displaystyle \mathbf {N} \!\,}\mathbf {N} \!\,
the (set of) natural numbers
N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.

The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analystsset theorists andcomputer scientists prefer the former. To avoid confusion, always check an author's definition of N.

Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤.
ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a ∈ ℤ}



R
{\displaystyle \mathbb {R} \!\,}\mathbb {R} \!\,

{\displaystyle \mathbf {R} \!\,}\mathbf {R} \!\,
R;
the (set of) real numbers;
the reals
ℝ means the set of real numbers.π ∈ ℝ

√(−1) ∉ ℝ

{\displaystyle \delta \!\,}\delta \!\,
Dirac delta of
{\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}\delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}δ(x)
Kronecker delta of
{\displaystyle \delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}\delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}δij
Functional derivative of
{\displaystyle {\begin{aligned}\left\langle {\frac {\delta F[\varphi (x)]}{\delta \varphi (x)}},f(x)\right\rangle &=\int {\frac {\delta F[\varphi (x)]}{\delta \varphi (x')}}f(x')dx'\\&=\lim _{\varepsilon \to 0}{\frac {F[\varphi (x)+\varepsilon f(x)]-F[\varphi (x)]}{\varepsilon }}\\&=\left.{\frac {d}{d\epsilon }}F[\varphi +\epsilon f]\right|_{\epsilon =0}.\end{aligned}}}{\begin{aligned}\left\langle {\frac {\delta F[\varphi (x)]}{\delta \varphi (x)}},f(x)\right\rangle &=\int {\frac {\delta F[\varphi (x)]}{\delta \varphi (x')}}f(x')dx'\\&=\lim _{\varepsilon \to 0}{\frac {F[\varphi (x)+\varepsilon f(x)]-F[\varphi (x)]}{\varepsilon }}\\&=\left.{\frac {d}{d\epsilon }}F[\varphi +\epsilon f]\right|_{\epsilon =0}.\end{aligned}}{\displaystyle {\frac {\delta V(r)}{\delta \rho (r')}}={\frac {1}{4\pi \epsilon _{0}|r-r'|}}}{\frac {\delta V(r)}{\delta \rho (r')}}={\frac {1}{4\pi \epsilon _{0}|r-r'|}}



{\displaystyle \vartriangle \!\,}\vartriangle \!\,

{\displaystyle \ominus \!\,}\ominus \!\,
symmetric difference
A ∆ B (or A ⊖ B) means the set of elements in exactly one of A or B.

(Not to be confused with delta, Δ, described below.)
{1,5,6,8} ∆ {2,5,8} = {1,2,6}

{3,4,5,6} ⊖ {1,2,5,6} = {1,2,3,4}

{\displaystyle \Delta \!\,}\Delta \!\,
delta;
change in
Δx means a (non-infinitesimal) change in x.

(If the change becomes infinitesimal, δ and even d are used instead. Not to be confused with the symmetric difference, written ∆, above.)
{\displaystyle {\tfrac {\Delta y}{\Delta x}}}{\tfrac {\Delta y}{\Delta x}} is the gradient of a straight line.
Laplace operator
The Laplace operator is a second order differential operator in n-dimensional Euclidean spaceIf ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒis defined by {\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}\Delta f=\nabla ^{2}f=\nabla \cdot \nabla f

{\displaystyle \sigma \!\,}\sigma \!\,
Selection of
The selection {\displaystyle \sigma _{a\theta b}(R)}\sigma _{a\theta b}(R) selects all those tuples in {\displaystyle R}R for which {\displaystyle \theta }\theta  holds between the {\displaystyle a}a and the {\displaystyle b}b attribute. The selection {\displaystyle \sigma _{a\theta v}(R)}\sigma _{a\theta v}(R) selects all those tuples in {\displaystyle R}R for which {\displaystyle \theta }\theta holds between the {\displaystyle a}a attribute and the value {\displaystyle v}v.{\displaystyle \sigma _{\mathrm {Age} \geq 34}(\mathrm {Person} )}{\displaystyle \sigma _{\mathrm {Age} \geq 34}(\mathrm {Person} )}
{\displaystyle \sigma _{\mathrm {Age} =\mathrm {Weight} }(\mathrm {Person} )}{\displaystyle \sigma _{\mathrm {Age} =\mathrm {Weight} }(\mathrm {Person} )}
{\displaystyle \sum }\sum
sum over ... from ... to ... of
{\displaystyle \sum _{k=1}^{n}{a_{k}}}\sum _{k=1}^{n}{a_{k}} means {\displaystyle a_{1}+a_{2}+\cdots +a_{n}}a_{1}+a_{2}+\cdots +a_{n}.{\displaystyle \sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30}\sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30


{ }
{\displaystyle \emptyset \!\,}\emptyset \!\,

{\displaystyle \varnothing \!\,}\varnothing \!\,

{\displaystyle \{\}\!\,}\{\}\!\,
the empty set
∅ means the set with no elements.[7] { } means the same.{n ∈ ℕ : 1 < n2 < 4} = ∅









{\displaystyle \blacksquare \!\,}\blacksquare \!\,

{\displaystyle \Box \!\,}\Box \!\,

{\displaystyle \blacktriangleright \!\,}\blacktriangleright \!\,
QED;
tombstone;
Halmos finality symbol
everywhere
Used to mark the end of a proof.

(May also be written Q.E.D.)
(1) a + 0 := a   (def.)

(2) a + succ(b) := succ(a + b)   (def.)

Proposition. 3 + 2 = 5.

Proof.

3 + 2 = 3 + succ(1)   (definition of succ)
3 + succ(1) = succ(3 + 1)   (2)
succ(3 + 1) = succ(3 + succ(0))   (definition of succ)
succ(3 + succ(0)) = succ(succ(3 + 0))   (2)
succ(succ(3 + 0)) = succ(succ(3))   (1)
succ(succ(3)) = succ(4) = 5   (definition of succ) 


{\displaystyle \approx }\approx
approximately equal
is approximately equal to
everywhere
x ≈ y means x is approximately equal to y.

This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒.
π ≈ 3.14159
is isomorphic to
G ≈ H means that group G is isomorphic (structurally identical) to group H.

(≅ can also be used for isomorphic, as described below.)
Q8 / C2 ≈ V

{\displaystyle \leq \!\,}\leq \!\,

{\displaystyle \geq }\geq
is less than or equal to,
is greater than or equal to
x ≤ y means x is less than or equal to y.

x ≥ y means x is greater than or equal to y.

(The forms <= and >= are generally used in programming languages, where ease of typing and use of ASCII text is preferred.)
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is a subgroup of
H ≤ G means H is a subgroup of G.Z ≤ Z
A3 ≤ S3
is reducible to
A ≤ B means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction.If
{\displaystyle \exists f\in F{\mbox{ . }}\forall x\in \mathbb {N} {\mbox{ . }}x\in A\Leftrightarrow f(x)\in B}\exists f\in F{\mbox{ . }}\forall x\in \mathbb {N} {\mbox{ . }}x\in A\Leftrightarrow f(x)\in B

then

{\displaystyle A\leq _{F}B}A\leq _{F}B





{\displaystyle \Rightarrow \!\,}\Rightarrow \!\,

{\displaystyle \rightarrow \!\,}\rightarrow \!\,

{\displaystyle \supset \!\,}\supset \!\,
implies;
if ... then
A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.

(→ may mean the same as ⇒, or it may have the meaning forfunctions given below.)

(⊃ may mean the same as ⇒,[4] or it may have the meaning forsuperset given below.)
x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2).



로컬 디렉터리에 파이썬 모듈 인스톨 하기 linux

[ref: http://scicomp.stackexchange.com/questions/2987/what-is-the-simplest-way-to-do-a-user-local-install-of-a-python-package]

다운받은 모듈 디렉터리로 이동하고 아래를 입력
$python setup.py install --user

그러면, $HOME/.local 밑으로 빌드가 되는데, 아래를 입력해서 설치
easy_install --prefix=$HOME/.local/ 모듈이름 

그리고 path설정만 .bashrc에 $HOME/.local을 추가해주면 됨

난 이렇게 pip 설치를 했는데,

pip로 virtualenv 설치하려니 또 sudo권한때문에 안되었는데,

pip로 설치할때도 --user 옵션을 주니까 $HOME/.local/에 설치가 잘된다

graphviz prog..........

참고 사이트
http://stackoverflow.com/questions/590821/keyboard-friendly-light-weight-uml-modeling-tool
http://www.graphviz.org/
http://www.ffnn.nl/pages/articles/media/uml-diagrams-using-graphviz-dot.php
http://www.umlgraph.org/


uml그리기에는 graphviz만한 툴이 없는 것 같다.

Visio나 staruml 같은 WISIWIG툴로 그리는게 쉬워보일 수는 있지만,

그려놓은 uml을 관리하거나, 스크립트로 뭔가 추가 작업이 필요할때는 graphviz가 좋은 도구임에 틀림없다고 생각한다.

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