The definition of a subset has be explained in the preceding post. Here, we shall use the definition.
If a set have many elements, there will be many subsets of the set.
Given a set $A$ which is not empty, a set whose elements are all subsets of $A$ is called the power set of $A$.
It is a set and expressed $\mathcal{A}$ .
\[ \forall x\exists z \forall w[w\in z\leftrightarrow w\subset x] \]
It is axiom of power set. If you do not want to use the symbol $\subset$ ,
\[ \forall x\exists z \forall w[w\in z\leftrightarrow \forall y[y\in w\rightarrow y\in x ]] . \]
If $A=\left\{ x,y \right\}$ , then all subsets of $a$ is $\left\{ x \right\}, \left\{ y \right\},$ and $\left\{ x,y \right\}$ .
Therefore, $\mathcal{A}=\left\{ \phi, \left\{ x \right\}, \left\{ y \right\}, \left\{ x,y \right\} \right\}$ .
As you see, if the number of elements of a set is $n$ , the number of the elements of the power set is $2^n$ .
(including empty set $\phi$.)
As $\phi=0$ , the power set of $\left\{ 0,1 \right\}$ is $\left\{ 0,\left\{ 0,1 \right\} \right\}=\left\{ 0, \left\{ 1 \right\} \right\}$ .
The power set of $\phi$ is $\left\{ 0 \right\}$ ,
and the power set of the power set of $\phi$ is $\left\{ 0, \left\{ 0 \right\} \right\}=\left\{ 0, 1 \right\}$ .
2016/01/12
2016/01/07
coffee break 11 (happy new year)
Happy new year
Recently I am playing middle-earth:shadow of mordor goty on steam.
The game is action RPG with open world. However, RPG elements are a little bit.
The action style is based on combats of Batman and wall climbing and stealth kills of Assassin Creed
These are very fun and feel good.
Player's character death is usually occurred in an action game.
Retrying the scene of the game, as player's skill becomes good, players can advance the game.
Then, there are normally no death penalties in action games.
However, there is a death penalty in this game and it is very interesting.
When player's character was slain, the enemies become level up and stronger.
Therefore, player's character can not afford to die.
Now as my character has been dead many times, I am afraid the clear of the game.
Recently I am playing middle-earth:shadow of mordor goty on steam.
The game is action RPG with open world. However, RPG elements are a little bit.
The action style is based on combats of Batman and wall climbing and stealth kills of Assassin Creed
These are very fun and feel good.
Player's character death is usually occurred in an action game.
Retrying the scene of the game, as player's skill becomes good, players can advance the game.
Then, there are normally no death penalties in action games.
However, there is a death penalty in this game and it is very interesting.
When player's character was slain, the enemies become level up and stronger.
Therefore, player's character can not afford to die.
Now as my character has been dead many times, I am afraid the clear of the game.