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Logical distinction between sets and algebraic structures

  1. Jan 1, 2010 #1
    Let's say we have a set S, and a function f : S -> S. Now let S be endowed with a binary operation, forming a group G. Is it correct to write f : G - > G?

    Up to now I have been operating on the assumption that yes, although G is not technically a set, there is little harm in being sloppy and use G to designate its underlying set, S.

    However someone has recently told me that this is not correct. f : G - > G is different from f : S - > S. I was referred to category theory, of which I admittedly know nothing.

    Is this true?
  2. jcsd
  3. Jan 1, 2010 #2


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    There are a variety of syntactic conventions... but I expect you were told something more conceptual: a map of groups really ought to be a homomorphism. Only certain set functions S -> S correspond to group homomorphisms G -> G.
  4. Jan 2, 2010 #3


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    A set is is the generic term for a collection of things. Group members, vectors, probability events, etc. are all elements of sets.
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