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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:

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 \} } \}

\)

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

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

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 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

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