Recently, I watched several movies below at a time.
Nodame Cantabile (She plays 'cantabile')
Minna ESPER dayo (Everyone is ESPERs)
Umimachi diary (Our little sister)
Biri gal（With flying colors）
Himizu (Mole)
Mamma mia
American Sniper
Star Wars ep.7 The force awakens
Watashi no otoko (My man)
Jigoku de naze warui (Why don't you play in hell?)
All is good and very interesting.
No.1 is 'Our little sister'.
The film was awarded the Japan Academy Award.
Its original story is the comic book and not finished.
The contents of the movie have about 5/7 on the published comic books.
'Umimachi' means a town nearby the shore in japanese language.
The movie makes me feel the atmosphere of the town and
the sea breeze comfortably.
Star Wars ep.7 disappointed me a little than old films.
2016/03/30
2016/03/02
axiomatic sets 12 (relations)
In axiomatic set theory, a relation is also a set.
Ordinally we will look at what is a relation between the element $x$ and $y$ .
For instance, in real numbers an ordered relation is satisfied.
Namely, in an arbitrary pair $\lt x,y\gt$ of two elements of real numbers
only one of the three relations
\[ x=y,\quad x\lt y,\quad x\gt y \]
must occure.
However, in axiomatic set theory we are interested in
the set $R$ which all $\left\{ \lt x,y\gt \right\}$ belongs to.
Please note that there is a set before a relation.
If $R$ is a binary relation,
\[ \forall z[z\in R\rightarrow \exists x\exists y[z=\lt x,y\gt ] ] . \]
,where $\lt x,y\gt$ is an ordered pair, which has been explained in the preceding post.
We will state it $xRy$ . $ xRy$ if and only if $\lt x,y\gt\in R$ .
If $R$ is a ternary relation,
\[ \forall z [z\in R\rightarrow \exists x\exists y\exists w[z=\lt\lt x,y\gt,w\gt ] ] . \]
In addition, well-known properties of a relation are as follow;
$R$ is reflexive in $A$ if and only if $\forall z[z\in A\rightarrow zRz ]$ .
$R$ is symmetric in $A$ if and only if $\forall x\forall y[x,y\in A , xRy \rightarrow yRx] $
$R$ is transitive in $A$ if and only if $\forall x\forall y\forall z[x,y,z\in A , xRy , yRz\rightarrow xRz] $
You must know the relation which satisfies above three conditions is an equivalence relation "=" on the set.
(This is not an axiom.)
Ordinally we will look at what is a relation between the element $x$ and $y$ .
For instance, in real numbers an ordered relation is satisfied.
Namely, in an arbitrary pair $\lt x,y\gt$ of two elements of real numbers
only one of the three relations
\[ x=y,\quad x\lt y,\quad x\gt y \]
must occure.
However, in axiomatic set theory we are interested in
the set $R$ which all $\left\{ \lt x,y\gt \right\}$ belongs to.
Please note that there is a set before a relation.
If $R$ is a binary relation,
\[ \forall z[z\in R\rightarrow \exists x\exists y[z=\lt x,y\gt ] ] . \]
,where $\lt x,y\gt$ is an ordered pair, which has been explained in the preceding post.
We will state it $xRy$ . $ xRy$ if and only if $\lt x,y\gt\in R$ .
If $R$ is a ternary relation,
\[ \forall z [z\in R\rightarrow \exists x\exists y\exists w[z=\lt\lt x,y\gt,w\gt ] ] . \]
In addition, well-known properties of a relation are as follow;
$R$ is reflexive in $A$ if and only if $\forall z[z\in A\rightarrow zRz ]$ .
$R$ is symmetric in $A$ if and only if $\forall x\forall y[x,y\in A , xRy \rightarrow yRx] $
$R$ is transitive in $A$ if and only if $\forall x\forall y\forall z[x,y,z\in A , xRy , yRz\rightarrow xRz] $
You must know the relation which satisfies above three conditions is an equivalence relation "=" on the set.
(This is not an axiom.)