Cardinality of Functions

In summary, the cardinality of a function refers to the number of elements in its domain and can be determined by counting or using mathematical notation. It can be infinite and can affect the function's properties, such as injectivity, surjectivity, and bijectivity. There may also be a relationship between the cardinality of the function and its range.
  • #1
moo5003
207
0
Is it true that the set of permutations on a infinite set K has the same cardinality as all functions between a infinite set K to itself?
 
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  • #2
Did you try to set up a bijection? Or did you try anything to (dis)prove what you're aksing for?
 
  • #3
To add to what Pere Callahan said: How many permutations are there on K= {a, b}?
How many functions are there from K to K?
(Permutations are one-to-one and onto functions.)
 

1. What is the cardinality of a function?

The cardinality of a function is the number of elements in the function's domain. It represents the size or amount of inputs that the function can take.

2. How is the cardinality of a function determined?

The cardinality of a function is determined by counting the number of elements in the function's domain. This can be done by listing out all the inputs or using mathematical notation to represent the size of the domain.

3. Can the cardinality of a function be infinite?

Yes, the cardinality of a function can be infinite. This means that the function has an uncountable number of elements in its domain, such as in the case of real numbers or continuous functions.

4. How does the cardinality of a function affect its properties?

The cardinality of a function can affect its properties in terms of its injectivity, surjectivity, and bijectivity. For example, a function with a smaller cardinality in its domain may have more restrictions and limitations compared to a function with a larger cardinality in its domain.

5. Is there a relationship between the cardinality of a function and its range?

Yes, there can be a relationship between the cardinality of a function and its range. For example, in some cases, a function may have the same cardinality in its domain and range, while in other cases, the cardinality of the range may be smaller than the cardinality of the domain.

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