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

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In summary, Peter is reading the book "Discovering Modern Set Theory. I The Basics" by Winfried Just and Martin Weese and is currently focused on Chapter 1. He is working on Exercise 8, which involves Cartesian Products and indexed families of sets. He has some questions about the exercise and has provided his working and solutions so far. He also has some questions about how to demonstrate the one-to-one map from the Cartesian Product to the set A_phi times A_{phi}.
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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:

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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Peter said:
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

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## 1. What is an indexed family of sets?

An indexed family of sets is a collection of sets that are associated with a common indexing set. The indexing set can be any set, such as the natural numbers or the real numbers, and each element in the indexing set corresponds to a unique set in the family.

## 2. How is an indexed family of sets different from a normal set?

An indexed family of sets is different from a normal set in that it is a collection of sets, rather than just a single set. It also has an indexing set, which is not present in a normal set.

## 3. What is the purpose of Exercise 8 in "Indexed Families of Sets ... Just and Weese ... "?

The purpose of Exercise 8 is to practice working with indexed families of sets and to deepen understanding of their properties and relationships. It involves applying concepts such as unions, intersections, and complements to indexed families of sets.

## 4. Can indexed families of sets have an infinite number of sets?

Yes, indexed families of sets can have an infinite number of sets. The indexing set can be an infinite set, such as the natural numbers, and therefore the family will also have an infinite number of sets.

## 5. How are indexed families of sets used in mathematics?

Indexed families of sets are used in many areas of mathematics, including set theory, topology, and analysis. They are particularly useful for defining sequences and for studying the convergence of sequences. They are also used in the construction of mathematical structures, such as topological spaces and metric spaces.

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