Some nice people have tried answering my question, and now I've totally understood the concept. However, I recently came across a wonderful explanation to this topic, which I think will help the new comers having the same question. This explanation is from the book "Discrete mathematics and its applications" by Kenneth H. Rosen, pg 597, 598:
A computer network has data centers in Boston, Chicago, Denver,
Detroit, New York, and San Diego. There are direct, one-way telephone
lines from Boston to Chicago, from Boston to Detroit, from Chicago to
Detroit, from Detroit to Denver, and from New York to San Diego. Let R
be the relation containing (a, b) if there is a telephone line from
the data center in a to that in b. How can we determine if there is
some (possibly indirect) link composed of one or more telephone lines
from one center to another? Because not all links are direct, such as
the link from Boston to Denver that goes through Detroit, R cannot be
used directly to answer this. In the language of relations, R is not
transitive, so it does not contain all the pairs that can be linked.
As we will show in this section, we can find all pairs of data centers
that have a link by constructing a transitive relation S containing R
such that S is a subset of every transitive relation containing R.
Here, S is the smallest transitive relation that contains R. This
relation is called the transitive closure of R.
In general, let R be a relation on a set A. R may or may not have some
property P, such as reflexivity, symmetry, or transitivity. If there
is a relation S with property P containing R such that S is a subset
of every relation with property P containing R, then S is called the
closure of R with respect to P. (Note that the closure of a relation
with respect to a property may not exist; see Exercises 15 and 35.)We
will show how reflexive, symmetric, and transitive closures of
relations can be found.
The relation R = {(1, 1), (1, 2), (2, 1), (3, 2)} on the set A = {1,
2, 3} is not reflexive. How can we produce a reflexive relation
containing R that is as small as possible? This can be done by adding
(2, 2) and (3, 3) to R, because these are the only pairs of the form
(a, a) that are not in R. Clearly, this new relation contains R.
Furthermore, any reflexive relation that contains R must also contain
(2, 2) and (3, 3). Because this relation contains R, is reflexive, and
is contained within every reflexive relation that contains R, it is
called the reflexive closure of R.
