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I Relations & functions

  1. Jul 25, 2016 #1
    Hello every one .
    A relation ( is a subset of the cartesian product between Xand Y) in math between two sets has spacial
    types 1-left unique ( injective)
    2- right unique ( functional )
    3- left total
    4- right total (surjective)
    May question is 1- a function ( map ) is a relation that is
    a- right unique
    b- left total
    I'm asking if there is a relation ( not function ) that is ( left total) and ( right total ) then what would is be called ? In the sense that the two set are infinite set is there and example
    My second question if we have two group structures and we want a relation between them , why does always the relation is function ( homomorphism ) ? Is there a relation that is left total and right total between the two structures ?
    Thanks
     
  2. jcsd
  3. Jul 25, 2016 #2

    fresh_42

    Staff: Mentor

    Does your x-total imply x-unique? If not, you have pretty many possibilities to define non-functional relations (finite or not).
    The same goes for group homomorphisms. Simply define a relation ##R## (functional or not, finite or not) with ##(a,1) \in R## for an ##a \neq 1##, the neutral element.
     
  4. Jul 25, 2016 #3
    No , there is no uniqueness
    A relation which is not function e.g. X^2+Y^2=1 , this is between two sets
    Now if a set with a structure ( space ) is there relation( not map ) between the two space or groups ? And how would it look like ?
     
  5. Jul 25, 2016 #4

    fresh_42

    Staff: Mentor

    Simply take a projection, e.g. ##ℝ^2 → ℝ## with ##(x,y) = x## and turn the arrow, so ##((x,y),x)## becomes ##(x,(x,y))##.
    But this is only one example out of many. Relation means, you are not restricted to any other rule than to draw many arrows, i.e. in case of totality ##R \subseteq X \times Y## such that ##∀ x \in X \; ∀ y \in Y \; ∃ (x,y) \in R##. Relate whatever you want to.
    There is a reason why we talk about functions. Relations are simply too many and too arbitrary.
     
  6. Jul 25, 2016 #5
    Thank you very much , now i understand why we choose functions to relate spaces , and alao i think functions appear in nature of physics alot ( by means function decribe the nature ) and easy to handle because we know how elements are related .
     
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