Function Definition Without a Single Word

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SUMMARY

The forum discussion centers on the correct definition of a function, specifically the notation used in set-builder form. The initial definition proposed, ψ = { (x, y) | ∀x∈A∃y∈B∋(((x, y) ∈ ψ) ∧((x,z)∈ ψ) ⇒ y = z)}, is criticized for being circular and lacking clarity regarding the elements of the set. Participants emphasize the necessity of specifying the domain and codomain in function definitions, as well as the importance of avoiding circular references. A corrected definition is suggested, F = {(x,y): (x,y) in AxB and if (x1,y1) is in AxB and x=x1 then y=y1}, which aims to clarify the relationship between sets A and B.

PREREQUISITES
  • Understanding of set-builder notation
  • Familiarity with the concepts of domain and codomain in functions
  • Knowledge of Cartesian products in set theory
  • Basic principles of mathematical logic
NEXT STEPS
  • Study the formal definition of functions in set theory
  • Learn about the implications of circular definitions in mathematics
  • Explore examples of function definitions using set-builder notation
  • Investigate the differences between total and partial functions
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Mathematicians, computer scientists, educators, and students seeking to deepen their understanding of function definitions and set theory.

  • #31
dijkarte said:
Interesting approach. So there's one and only one universe that contains all possible mappings from sets A to B, and any f:A --> B is an element of this. I will research the topic further.

Well it's a set. You can get a handle on this by working out a simple example with small finite sets. For example let

A = {a1, a2, a3} and B = {b1, b2}. How many functions can there be from A to B?

Well a1 can go to b1 or b2.

a2 can go to b1 or b2.

a3 can go to b1 or b2.

So that gives us 2x2x2 = 8 possible functions from A to B. Each function is a set of exactly three ordered pairs, and there are only eight functions. Not too hard to write them all down.

Now you can see why we chose the notation B^A for the set of all functions from A to B. Because the set B^A has cardinality exactly |B|^|A|.
 

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