Relations & Functions: Types, Examples, Homomorphism

[mikeeey]

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

 

[fresh_42]

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.

 

[mikeeey]

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 ?

 

[fresh_42]

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.

 

[mikeeey]

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 .