What is the difference between a relation and a closure?

Question

I know what a transitive, reflexive and symmetric relation is. When I study transitive, reflexive and symmetric closure of a binary relation, I find it difficult to get an intuition and so am unable to differentiate it with their corresponding relations. So can anyone please give an analogy kind of thing to make it easy to understand "closure" and relation between a "relation" and its "closure"?

Today is the first day I've come across these terms so apologies for this silly question (if it really is).

Answer 1

If you have a relation $R$, its transitive closure $R^+$ is the smallest transitive relation such that $R \subseteq R^+$. If $R$ is already transitive, then $R = R^+$. So a transitive closure is also a relation, and it is the relation that is obtained by expanding the original relation in such a way as to make it transitive. The same idea applies to reflexive closure and symmetric closure.

Answer 2

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.