what are transitivity reflexive symmetry, Reflexive closure? 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..
 A: 
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 $R=\{(1,1), (1,3), (3,2), (3,3)\}
\\ R^2 =\{(1,1), (1,2), (1,3), (3,2), (3,3)\} = R^3$
$$R^+~=~\{(1,1), (1,2), (1,3), (3,2), (3,3)\}$$
