Supremum And Infimum Of The Product Of A Family Of Sets Of Cardinals

Introduction

In the realm of set theory, the concepts of supremum and infimum play a crucial role in understanding the properties of sets, particularly when dealing with cardinal numbers. The product of a family of sets of cardinals is a fundamental concept that has far-reaching implications in various areas of mathematics. In this article, we will delve into the relationship between the supremum and infimum of the product of a family of sets of cardinals, exploring the validity of the inequality:

$$\prod \sup A_i\leq \sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\}$$

Preliminaries

Before we embark on our investigation, let us establish some essential definitions and notations.

• A cardinal number is a measure of the size of a set, which can be thought of as the number of elements in the set.
• The supremum of a set of cardinal numbers, denoted by $\sup A$, is the smallest cardinal number that is greater than or equal to every cardinal number in the set.
• The infimum of a set of cardinal numbers, denoted by $\inf A$, is the largest cardinal number that is less than or equal to every cardinal number in the set.
• The product of a family of sets of cardinals, denoted by $\prod A_i$, is the cardinal number that represents the size of the Cartesian product of the sets $A_i$.

The Inequality

We are interested in determining whether the following inequality holds:

$$\prod \sup A_i\leq \sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\}$$

To approach this problem, let us consider a specific example. Suppose we have a family of sets of cardinals $\{A_i\}_{i\in I}$, where each set $A_i$ contains a single cardinal number $a_i$. In this case, the product of the sets is simply the product of the cardinal numbers:

$$\prod A_i = \prod a_i$$

Now, let us examine the supremum of the product of the sets:

$$\prod \sup A_i = \prod \sup \{a_i\} = \prod a_i$$

On the other hand, the supremum of the product of the cardinal numbers is:

$$\sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\} = \prod a_i$$

In this specific example, we can see that the inequality holds:

$$\prod \sup A_i = \prod a_i \leq \prod a_i = \sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\}$$

However, this example is highly contrived, and we need to consider more general cases to determine whether the inequality holds in general.

General Case

Let us consider a more general case, where each set $A_i$ contains multiple cardinal numbers. In this case, the product of the sets is the product of the cardinal numbers:

$$\prod A_i = \prod \{a_{ij}\}$$

where $a_{ij}$ is a cardinal number in the set $A_i$. The supremum of the product of the sets is:

$$\prod \sup A_i = \prod \sup \{a_{ij}\}$$

On the other hand, the supremum of the product of the cardinal numbers is:

$$\sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\} = \sup \prod \{a_{ij}\}$$

In this general case, we can see that the inequality may not hold in general. For example, consider a family of sets of cardinals $\{A_i\}_{i\in I}$, where each set $A_i$ contains two cardinal numbers $a_{i1}$ and $a_{i2}$. In this case, the product of the sets is:

$$\prod A_i = \prod \{a_{i1}, a_{i2}\}$$

The supremum of the product of the sets is:

$$\prod \sup A_i = \prod \sup \{a_{i1}, a_{i2}\} = \prod (a_{i1} + a_{i2})$$

On the other hand, the supremum of the product of the cardinal numbers is:

$$\sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\} = \sup \prod \{a_{i1}, a_{i2}\}$$

In this case, the inequality does not hold:

$$\prod \sup A_i = \prod (a_{i1} + a_{i2}) \not\leq \sup \prod \{a_{i1}, a_{i2}\}$$

Conclusion

In conclusion, we have investigated the relationship between the supremum and infimum of the product of a family of sets of cardinals. We have shown that the inequality:

$$\prod \sup A_i\leq \sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\}$$

does not hold in general. However, we have also shown that the inequality holds in specific cases, such as when each set $A_i$ contains a single cardinal number. Further research is needed to determine the conditions under which the inequality holds.

References

• [1] Jech, T. (2003). Set theory. Springer.
• [2] Kunen, K. (1980). Set theory: an introduction to independence proofs. North-Holland.
• [3] Devlin, K. J. (1973). Aspects of constructibility. Springer.

Future Work

• Investigate the conditions under which the inequality holds.
• Explore the relationship between the supremum and infimum of the product of a family of sets of cardinals and other mathematical concepts, such as ordinals and products.
• Develop new techniques for calculating the supremum and infimum of the product of a family of sets of cardinals.
  Supremum and Infimum of the Product of a Family of Sets of Cardinals: Q&A ====================================================================

Introduction

In our previous article, we explored the relationship between the supremum and infimum of the product of a family of sets of cardinals. We investigated the inequality:

$$\prod \sup A_i\leq \sup\Big\{ \prod a_i : a_i\in A_i; \mbox{ for all } i\in I\Big\}$$

and found that it does not hold in general. However, we also showed that the inequality holds in specific cases. In this article, we will answer some frequently asked questions (FAQs) related to the supremum and infimum of the product of a family of sets of cardinals.

Q: What is the product of a family of sets of cardinals?

A: The product of a family of sets of cardinals is the cardinal number that represents the size of the Cartesian product of the sets. In other words, it is the cardinal number that represents the number of ways to choose one element from each set in the family.

Q: What is the supremum of a set of cardinal numbers?

A: The supremum of a set of cardinal numbers is the smallest cardinal number that is greater than or equal to every cardinal number in the set. In other words, it is the smallest cardinal number that is an upper bound for the set.

Q: What is the infimum of a set of cardinal numbers?

A: The infimum of a set of cardinal numbers is the largest cardinal number that is less than or equal to every cardinal number in the set. In other words, it is the largest cardinal number that is a lower bound for the set.

Q: Why is the inequality not true in general?

A: The inequality is not true in general because the supremum of the product of the sets is not necessarily less than or equal to the supremum of the product of the cardinal numbers. This is because the product of the sets can be much larger than the product of the cardinal numbers.

Q: When does the inequality hold?

A: The inequality holds in specific cases, such as when each set $A_i$ contains a single cardinal number. In this case, the product of the sets is simply the product of the cardinal numbers, and the inequality holds.

Q: What are some applications of the supremum and infimum of the product of a family of sets of cardinals?

A: The supremum and infimum of the product of a family of sets of cardinals have applications in various areas of mathematics, such as:

• Set theory: The supremum and infimum of the product of a family of sets of cardinals are used to study the properties of sets and their cardinalities.
• Measure theory: The supremum and infimum of the product of a family of sets of cardinals are used to study the properties of measures and their relationships to sets.
• Topology: The supremum and infimum of the product of a family of sets of cardinals are used to study the properties of topological spaces and their relationships to sets.

Q: What are some open problems related to the supremum and infimum of the product of a family of sets of cardinals?

A: Some open problems related to the supremum and infimum of the product of a family of sets of cardinals include:

• Determining the conditions under which the inequality holds: Researchers are still working to determine the conditions under which the inequality holds.
• Developing new techniques for calculating the supremum and infimum of the product of a family of sets of cardinals: Researchers are still working to develop new techniques for calculating the supremum and infimum of the product of a family of sets of cardinals.
• Exploring the relationships between the supremum and infimum of the product of a family of sets of cardinals and other mathematical concepts: Researchers are still working to explore the relationships between the supremum and infimum of the product of a family of sets of cardinals and other mathematical concepts, such as ordinals and products.

Conclusion

In conclusion, the supremum and infimum of the product of a family of sets of cardinals are fundamental concepts in mathematics that have far-reaching implications in various areas of mathematics. We have answered some frequently asked questions related to these concepts and highlighted some open problems that researchers are still working to solve. Further research is needed to fully understand the properties and relationships of these concepts.