I have seen these and honestly I did not understand what does they do, what is useful or important about them? I always confused at seeing them.
can someone point a source with good real world examples or give some explanation here? what is the Reflexive closure? with some examples..
what is the Reflexive closure? with some examples..
The Reflexive Closure of a relation $R$ is the smallest reflexive relation that contains $R$. Basically it is $R$ plus any ordered pair necessary to have a reflexive relation. A reflexive relation is, of course, its own closure.
If $A$ is the set and $R$ is a relation on that set ($R\subseteq A^2$) then its reflexive closure, $S$, is : $$S= R\cup\{(x,x): x\in A\}$$
Example 1:
Let $A=\{1,2,3\}$ and $R=\{(1,1), (1,3), (3,2), (3,3)\}$. $R$ is a binary relation on $A$.
$R$ is not reflexive because it does not contain the pair $(2,2)$. The reflexive closure is thus: $$S=\{(1,1), (1,3), (2,2), (3,2), (3,3)\}$$
The Transitive and Symmetric closures are similarly defined.
The Symmetric Closure of a relation is the smallest symmetric relation which contains the relation. Basically it adds any missing mirror elements. In other words, it is the union of the relation and its inverse.
$$R\cup R^{-1}~=~ R\cup \{(y,x): (x,y)\in R\}$$
Example 2 Using the same set and relation as above. $R$ is not symmetric because $(1,3)$ and $(3,2)$ do not have inverses. $R^{-1}~=~\{(1,1), (3,1), (2,3), (3,3)\}$
$$R\cup R^{-1} ~=~ \{(1,1), (1,3), (2,3), (3,1), (3,2), (3,3)\}$$
The Transitive Closure of a Relation is the smallest transitive relation which contains the relation. Basically we add the necessary transitive "bridge" elements.
This is much harder to express as a set constructor. It is defined as the union series of all composite powers of $R$ $$R^+ ~=~ \bigcup_{i\in\{1,2,...\}} R^{i}$$
Where we recursively define $R^1 \mathop{:=} R, \forall i\in\{1,2,...\}~\big[ R^{i+1} := R\circ R^{i}\big]$
Composition is, $U\circ V=\{(v,u)\mid \exists w:(v,w)\in V\wedge (w,u)\in U\}$
Example 3:
Since $R=\{(1,1), (1,3), (3,2), (3,3)\} \\ R^2 =\{(1,1), (1,2), (1,3), (3,2), (3,3)\} = R^3$
$$R^+~=~\{(1,1), (1,2), (1,3), (3,2), (3,3)\}$$
