difficult question

KerpetenAli

New member
Joined
Jun 28, 2020
Messages
2
WhatsApp Image 2020-06-28 at 20.05.48 (1).jpeg
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
9,979
Here is your post: \(x,y\) aren't empty, infinity series and \(x\cap y=\emptyset\).
Show us \(f(x\cup y)\approx f(x)\times f(y)\).
That makes no sense whatsoever. You need to post a well formed question in English.
You must explain the terms used and the object of the question.
 

KerpetenAli

New member
Joined
Jun 28, 2020
Messages
2
I'm sorry I couldn't explain the question correctly. Original question: (P= power set) IMG-20200628-WA0037.jpg
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
7,374
So what have you tried, where are you stuck? What does P( X U Y) mean? How about P(X)xP(Y)?
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
9,979
I'm sorry I couldn't explain the question correctly. Original question: (P= power set)
Well that is somewhat better but only by a bit.
In the LaTeX markup language \(\approx\) means approximately.
Moreover, in almost all western mathematics if \(A~\&~B\) are sets, the notation \(A\times B\) is the cross product of the two sets.
As posted that makes no sense. Please define all your terms.
If necessary go to some standard website such as this or even this one.
Type in key words to find equivalent standard western notation.
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
8,037
I'm sorry I couldn't explain the question correctly. Original question: (P= power set)
I presume you are to show that \(\displaystyle P(X\cup Y)\approx P(X)\times P(Y)\) in some sense, saying that the power set of the union is ... what? ... to the Cartesian product of the individual power sets. What does "\(\displaystyle \approx\)" mean to you here? Please define it. I can see a couple kinds of relationship that would apply, but not "approximately equal".
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
9,979
I'm sorry I couldn't explain the question correctly. Original question: (P= power set)
This is a dangerous thing to do: I am going to guess that you are asked to show that
\(\mathscr{P}(X\cup Y)\) equipotent with \(\mathscr{P}(X)\times\mathscr{P}(Y)\). [Those are power sets.]
Sets \(U~\&~V\) are equipotent iff there are injections \(U\to V~\&~V\to U\).
If \(A\in\mathscr{P}(X\cup Y)\) then \(A\subseteq X\cup Y\). Let us define \(A_X=A\cap X~\&~A_Y=A\cap Y\).
Can you argue that \(A_X\cap A_Y=\emptyset~?\)
Can you show that \(\Phi(A)=(A_X,A_Y)\) is an injection \(\mathscr{P}(X\cup Y)\to\mathscr{P}(X)\times\mathscr{P}(Y)\)?
Now can YOU find an injection \(\mathscr{P}(X)\times\mathscr{P}(Y) \to \mathscr{P}(X\cup Y)~?\)
 
Top