I Relations & Functions: Types, Examples, Homomorphism

AI Thread Summary
A relation in mathematics is defined as a subset of the Cartesian product between two sets, with types including injective, functional, left total, and surjective. A function is a specific type of relation that is both right unique and left total. The discussion raises questions about the existence of relations that are left total and right total but not functions, particularly in the context of infinite sets. It also explores why relations between group structures often result in functions, or homomorphisms, emphasizing that relations can be arbitrary and numerous. The conclusion highlights the preference for functions in mathematical contexts due to their clear and manageable nature, especially in describing natural phenomena.
mikeeey
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Hello every one .
A relation ( is a subset of the cartesian product between Xand Y) in math between two sets has spatial
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
 
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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.
 
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 ?
 
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.
 
Thank you very much , now i understand why we choose functions to relate spaces , and alao i think functions appear in nature of physics a lot ( by means function decribe the nature ) and easy to handle because we know how elements are related .
 
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