1/31/15

Binary Relation & Stitie Space.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A.

More generally, a binary relation between two sets A and B is a subset of A × B.

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X Y. The sets X and Y are called the domain (or the set of departure) and codomain (or the set of destination), respectively, of the relation, and G is called its graph.


Example:

we have triple (X, Y, G) with x0 ∈ X, y0 ∈ Y;

if (x0, y0) ∈ G, then:

r0 = {x0 ,y0}.

we can say that (x0, y0) are in a relation r0 ∈ G.


Function:

we can also define binary relation with a function f: X × Y → B, G ⊂ X × Y.

where B is a set of boolean values: B = { true, false }.

(x,y) ∈ X × Y: ( ( f(x,y) = true ) ⇔ ( (x,y) ∈ G ) ) ∧ ( ( f(x,y) = false ) ⇔ ( (x,y) ∉ G ) ).


this is read:

we have a set of pairs (x,y) in a cartesian product X × Y that:

  f(x,y) has value of 'truth' if & only if (x,y) is in G,
&
  f(x,y) has value of 'falsehood' if & only if (x,y) is not in G,


we consider all pairs (x,y) ∈ X × Y.

pair (x,y) is in relation if & only if f(x,y) has value of 'truth'.


Uses:

- Stitie Space can be used to model & visualize relations graphs. we can have even discrete logic that way, with different 'truth objects' possibly forming inheritance tree. falsehood would be lack of any 'truth object' at given coordinates then. 'truths' could differ, but also could have something in common - could be the same or different subclasses of 'supertruth(s)'. this is not the same as 'fuzzy logic' that i understand as joining probablity calculus with boolean algebra, perhaps more.

- Certain types of Binary Relations can be used to determine ordering of elements in data structures.


See also, if You wish: What is a Function?, Ordering a Set, 'Ola AH' Programming Language.

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