Indexed Families of Sets .... Just and Weese .... Exercise 8 ....

[Math Amateur]

I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese.

I am currently focused on Chapter 1 Pairs, Relations and Functions ... and I am in particular focused on Cartesian Products and indexed families of sets ...

I need some help with Exercise 8 and some remarks following the exercise ...

The relevant section from J&W is as follows:View attachment 7533
View attachment 7534

It is also worth noting that earlier (on page 12) J&W defined ordered pairs and Cartesian Products as follows:

https://www.physicsforums.com/attachments/7535

I worked Exercise 8 as follows:

Elements of the set \(\displaystyle A_\phi \times A_{ \{ \phi \} }\)


Now ... \(\displaystyle \{ A_\phi = \{ \phi \}\) and \(\displaystyle A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}\) ...So \(\displaystyle A_\phi \times A_{ \{ \phi \} } = \{ \langle a,b \rangle \ \mid \ a \in A_\phi \text{ and } b \in A_{ \{ \phi \} } \}
\)

\(\displaystyle = \{ \ \langle \phi , \phi \rangle \ , \langle \phi , \{ \phi \} \rangle \ \} \)

\(\displaystyle = \{ \ \{ \phi , \{ \phi \} \} \ , \ \{ \phi , \{ \{ \phi \} \} \ \} \)

Elements of the set \(\displaystyle \prod_{ i = \{ \phi , \{ \phi \} } A_i \)


Let \(\displaystyle I = \{ \phi , \{ \phi \} \} \)
\(\displaystyle \prod_{ i \in I } A_i = \{ f \in ( \bigcup \{ A_i \ : \ i \in I \} )^I \ : \ \forall i \in I , \ ( f(i) \in A_i ) \} \)

where \(\displaystyle ( \bigcup \{ A_i \ : \ i \in I \} )^I = \{ f \ : \ I \rightarrow \bigcup A_i \} \)

\(\displaystyle = \{ f \ : \ \{ \phi , \{ \phi \} \} \rightarrow \bigcup A_i \} \)
Now ... we have to consider the function(s) from the domain \(\displaystyle \{ \phi , \{ \phi \} \}\) to

\(\displaystyle \bigcup A_i = A_\phi \cup A_{ \{ \phi \} } \)

\(\displaystyle = \{ \phi , \{ \phi \} \}\)The only function satisfying the required conditions is the following function f:

\(\displaystyle f = \{ \langle \phi , \phi \rangle , \langle \{ \phi \} , \{ \phi \} \rangle \} \)

\(\displaystyle = \{ \{ \phi , \{ \phi \} \} , \{ \{ \phi \} , \{ \{ \phi \} \} \}\)
My questions as follows:

Can someone please either confirm my working as correct or point out the errors ...?

Further, can someone show me simply and explicitly how H is a one to one map from \(\displaystyle \prod_{ i = \{ \phi , \{ \phi \} \} } A_i \) onto \(\displaystyle A_\phi \times A_{ \{ \phi \} }\) ... ... ?

Help will be much appreciated ...

Peter

 

[Math Amateur]

I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese.

I am currently focused on Chapter 1 Pairs, Relations and Functions ... and I am in particular focused on Cartesian Products and indexed families of sets ...

I need some help with Exercise 8 and some remarks following the exercise ...

The relevant section from J&W is as follows:It is also worth noting that earlier (on page 12) J&W defined ordered pairs and Cartesian Products as follows:
I worked Exercise 8 as follows:

Elements of the set \(\displaystyle A_\phi \times A_{ \{ \phi \} }\)


Now ... \(\displaystyle \{ A_\phi = \{ \phi \}\) and \(\displaystyle A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}\) ...So \(\displaystyle A_\phi \times A_{ \{ \phi \} } = \{ \langle a,b \rangle \ \mid \ a \in A_\phi \text{ and } b \in A_{ \{ \phi \} } \}
\)

\(\displaystyle = \{ \ \langle \phi , \phi \rangle \ , \langle \phi , \{ \phi \} \rangle \ \} \)

\(\displaystyle = \{ \ \{ \phi , \{ \phi \} \} \ , \ \{ \phi , \{ \{ \phi \} \} \ \} \)

Elements of the set \(\displaystyle \prod_{ i = \{ \phi , \{ \phi \} } A_i \)


Let \(\displaystyle I = \{ \phi , \{ \phi \} \} \)
\(\displaystyle \prod_{ i \in I } A_i = \{ f \in ( \bigcup \{ A_i \ : \ i \in I \} )^I \ : \ \forall i \in I , \ ( f(i) \in A_i ) \} \)

where \(\displaystyle ( \bigcup \{ A_i \ : \ i \in I \} )^I = \{ f \ : \ I \rightarrow \bigcup A_i \} \)

\(\displaystyle = \{ f \ : \ \{ \phi , \{ \phi \} \} \rightarrow \bigcup A_i \} \)
Now ... we have to consider the function(s) from the domain \(\displaystyle \{ \phi , \{ \phi \} \}\) to

\(\displaystyle \bigcup A_i = A_\phi \cup A_{ \{ \phi \} } \)

\(\displaystyle = \{ \phi , \{ \phi \} \}\)The only function satisfying the required conditions is the following function f:

\(\displaystyle f = \{ \langle \phi , \phi \rangle , \langle \{ \phi \} , \{ \phi \} \rangle \} \)

\(\displaystyle = \{ \{ \phi , \{ \phi \} \} , \{ \{ \phi \} , \{ \{ \phi \} \} \}\)
My questions as follows:

Can someone please either confirm my working as correct or point out the errors ...?

Further, can someone show me simply and explicitly how H is a one to one map from \(\displaystyle \prod_{ i = \{ \phi , \{ \phi \} \} } A_i \) onto \(\displaystyle A_\phi \times A_{ \{ \phi \} }\) ... ... ?

Help will be much appreciated ...

Peter

I have been reflecting on my solution to Just and Weese Exercise 8 ...

I know think I may have made an error in determining the set \(\displaystyle \prod_{ i = \{ \phi , \{ \phi \} } A_i \)

... ... so I am now attempting to give a correct solution ...
We have that ... \(\displaystyle A_\phi = \{ \phi \}\) and \(\displaystyle A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}\) ...and we let \(\displaystyle I = \{ \phi , \{ \phi \} \} \)
Then ... ... \(\displaystyle {}^I \!\left( \bigcup \{ A_i \}\right) = \{ f \ : \ I \rightarrow \bigcup A_i \}\) But ... ... \(\displaystyle \bigcup A_i = \bigcup \{ A_\phi, A_{ \{ \phi \} } \} = A_\phi \cup A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}
\)Therefore ... \(\displaystyle {}^I \!\left( \bigcup \{ A_i \}\right) = \{ f \ : \ \{ \phi , \{ \phi \} \} \rightarrow \{ \phi , \{ \phi \} \}\) ... ... (1)and \(\displaystyle \prod_{ i \in I } A_i = \{ f \in {}^I \!\left( \bigcup \{ A_i \}\right) \ : \ \forall i \in I \ (f(i) \in A_i \ ) \ \} \) ... ... ... (2)So ... working from (1) and (2) we have that \(\displaystyle \prod_{ i \in I } A_i = \{ g, h \}\)where (treating functions as sets of ordered pairs ...)\(\displaystyle g = \{ \ \langle \phi , \phi \rangle \ , \ \langle \{ \phi \} , \phi \rangle \ , \} = \{ \ \{ \phi , \{ \phi \} \} \ , \{ \{ \phi \} , \{ \phi \} \} \}\)and \(\displaystyle h = = \{ \ \langle \phi , \phi \rangle \ , \ \langle \{ \phi \} , \{ \phi \} \rangle \ \} = \{ \ \{ \phi , \{ \phi \} \} \ , \{ \{ \phi \} , \{ \{ \phi \} \} \ \} \)... ... BUT ... ... where to from here ... hmmm ...really need some help/guidance ...In particular how exactly and explicitly do we demonstrate the one-to-one map H from \(\displaystyle \prod_{ i = \{ \phi , \{ \phi \} } A_i \) to \(\displaystyle A_\phi \times A_{ \{ \phi \} }\) ... .. Help will be much appreciated ...

Peter