Re: Cantor's definition of set

John Jones schrieb:
It's my night off so you are getting more than your fair share of
replies.
Q: How is a function like a set? We can call it a set, but why?

Welcome on planet math.

The first thing is, you have to be able to construct
pairs. There are different approaches to it to do
it in set theory, but in the end you will have
functions and a new set constructor x:

<,>: A -> (B -> A x B)
p1: A->(B->A)
p2: A->(B->B)

With:

forall x in A, forall y in B p1(<x,y>)=x
forall x in A, forall y in B p2(<x,y>)=y

Example:

<1,2> = the pair built from 1 and 2
p1(<1,2>) = 1, the first projection of this pair
p2(<1,2>) = 2, the second projection of this pair

For each set A, B. Or even for the whole domain
of discourse U.

From pairs, you can get relations. A relation R is
simply a set of pairs.

R subset A x B.

That a relation holds between two elements is written
R(x,y) and is essentially a shorthand for:

R(x,y) <-> <x,y> in R

Example:

R={<1,2>,<3,4>}
R(1,2) holds
R(4,3) doesn't hold

The concept of relations can be extended not n-ary
relations. A "set of cutlery" is a relation.

From relations one can get functions. A (total)
function f: A->B is simply a relation:

f subset A x B

With the following conditions:

existence:

forall x in A
exists y in B f(x,y)

uniqueness:

forall x in A
forall y1 in B
forall y2 in B (f(x,y1) & f(x,y2) -> y1=y2)

Example:
R1={<1,2>,<3,4>}
R2={<3,4>}
R3={<1,2>,<3,4>,<3,2>}
R1 is a function {1,3} -> {2,4}
R2 is not a (total) function {1,3} -> {2,4}
existence is violated
R3 is not a function {1,3} -> {2,4}
uniqueness is violated

The concept of functions can be extended not n-ary
functions.

Bye

.

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