Function Definition Without a Single Word

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The discussion centers around the correct definition of a function using set-builder notation. Participants argue that the initial definition provided is circular and lacks clarity since it references the set ψ before it is fully defined. They emphasize the need to specify the domain and codomain in a function's definition, highlighting that a function should not be defined in terms of itself. The conversation also touches on the importance of distinguishing between different functions that may share the same graph. Ultimately, the need for a clear and non-circular definition of a function is underscored, with suggestions for improvement offered throughout the exchange.
  • #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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