Are set theory functions sets too?

In summary, the conversation discusses the possibility of implementing mathematical functions as sets in set theory. It is suggested that functions with a set domain can be represented as sets, but it is noted that the power set function cannot be described as a set due to Russel's paradox.
  • #1
The UPC P
9
0
I read somewhere that mathematical functions can be implemented as sets by making a set of ordered tuples <a,b> where a is a member of A and b is a member of B. That should create a function that goes from the domain A to the range B.

But set theory has functions too, could they be sets too?

For example the Power function would just be the set witht eh tuples <{},{{}}> and <{{}},{{}{{}}}> and so on. And the union and the pair function could be made into sets as well.

So what I want to ask is can all functions in set theory be defined as sets themselves?
 
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  • #2
The UPC P said:
So what I want to ask is can all functions in set theory be defined as sets themselves?

Yes, if the domain of the function is a set. The power set function has as domain the class of all sets, this is not a set due to Russel's paradox. So the power set function can not be described as a set. Rather, it must be described as a logical formula.
 
  • #3
OK thanks!
 

FAQ: Are set theory functions sets too?

1. What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It provides a foundation for other areas of mathematics, including functions and relations.

2. What are functions in set theory?

In set theory, functions are a type of relation between sets, where each input element from one set is paired with exactly one output element from another set. They are used to describe the relationship between two sets and can be represented using ordered pairs.

3. Are set theory functions also sets?

Yes, set theory functions are also sets. This is because they are defined as a type of relation, which is a subset of the Cartesian product of two sets. Therefore, functions can be treated as sets and follow the same rules of set theory.

4. How are set theory functions different from sets?

While set theory functions are also sets, they have different properties and purposes. Sets are collections of distinct objects, while functions are used to describe the relationship between two sets. Functions also have specific characteristics, such as being well-defined and having a domain and range.

5. How are set theory functions used in science?

Set theory functions are used in various scientific fields, such as physics, biology, and computer science. They are used to model real-world relationships and can help in understanding complex systems and solving problems. For example, in biology, functions can be used to represent the relationship between genes and traits in genetics.

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