Relationships
Relations relate two or more sets
Given two sets X and Y, their Cartesian product is denoted as X x Y=\{(x, y)|x\in X, y\in Y\} where (x, y) denotes an ordered pair consisting of x and y (this Generalization of Cartesian product can be applied to more than two sets)
A subset XxY is called relationship
When we have \mathbb{R}\subset X x X a relation in X, where (a, b)\in \mathbb{R} denotes aRb. These relationships can perform the following properties:
- Property reflective when all the a \in X meet aRa
- The property of symmetric if aRb is met bRa must also be met (it can be abbreviated how a R b\Rightarrow b R a)
- Property transitive if aRb and bRc are met then it must be met that aRc (you can shortly denote how a R b, a R c\Rightarrow a R c)
A relation that satisfies these three properties is called relationship of equivalence. Instead of using R, to denote them it is used \sim
The simplest example of an equivalence relationship is the equality relationship (each element is related only to itself). And if in an equivalence relationship we identify the related elements, we will get a kind of equality
We assume that in X we have a relation of equivalence \sim, group each element a\in R with everyone who is related to him. In this way we get for each to the following set: \hat{a}=\{x \in X| a \sim x\}. This set is called class of equivalence of a
If we have two elements a, b\in X, their respective equivalence classes \hat{a} and \hat{b} are the same (\hat{a} =\hat{b}) or disjoint (\hat{a}\cap\hat{b} = \emptyset), the different types of equivalence form what is called partition of X (by definition, a partition in a set is a series of subsets that are two-to-two disjoints and whose join results in the set)
The set of equivalence classes is a new set that is named set quotient and is denoted: \displaystyle X\backslash\sim=\{\hat{a}|a\in X\}
If a, b \in X meet a \sim b, your equivalence classes will be \hat{a} = \hat{b} and therefore are the same element in X\backslash\sim
If the symmetric property is not met, the following can be true:
A reflective, antisymetric and transitive R relationship is called order relationship, and it is common to denote it by \leq; it is said that (X, \leq) is an orderly set. With the same meaning as a\leq b it is also used b\geq a; if in addition to a\leq b we want to make sure that a=b, you can use a < b \text{ o }b > a
In an orderly set, if it is always true that a\leq b\text{ o }b\leq a, is called total order; otherwise we are faced with a partial order
When S is an ordered subset of X, we say that x \in X it is a upper bound of S if x \geq t for any t \in S. If there is a, the smallest of the upper dimensions and is called supreme S; if the supreme is in S, it is said to be the maximum S
When S is an ordered subset of X, we say that x \in X it is a lower elevation of S if x\geq t for any \displaystyle t \in S. If there is a, the largest of the lower dimensions and is called tiny S; if the still is in S, it is said to be the minimum S
A set in a well-ordered (or that complies with the principle of good sorting), is such an orderly set that any non empty subset has minimal