I've been given the following function where:
$$R:0..10 \iff 0..10 $$
$$ R = \{ m,n\colon\mathbb{N} \| m \neq n \land m + n = 8 \} $$
Therefore, I've calculated R as:
$$ R = \{(0,8),(1,7),(2,6),(3,5),(5,3),(6,2),(7,1),(8,0)\} $$
Reflexive Closure:
$$\{(0,0),(1,1),(2,2),(3,3),(5,5),(6,6),(7,7),(8,8)\} $$
Symmetric Closure:
$$ \{(8,0),(7,1),(6,2),(5,3),(3,5),(2,6),(1,7),(0,8)\} $$
Transitive Closure:
$$ \{(3,3)\} $$
Are these correct?
Best Answer
For the reflexive closure, you have to add the points $(n,n)$ not only for $n$ in the field of your relation, but for $n$ in the set over which you are viewing your relation. Since the context here seems to be relations on the natural numbers between $0$ and $10$, according to what you wrote at first, then I would say that the reflexive closure of $R$ has all the elements of $R$ plus the pairs $(n,n)$ for $n\in\{0,1,\ldots,10\}$. So you need also to add $(4,4)$ and $(9,9)$ and $(10,10)$.
For the symmetric closure, what you wrote is fine, but actually it is the same as $R$ itself, since $R$ is already symmetric! So the symmetric closure of $R$ is $R$ itself.
For the transitive closure, you will end up adding the points that you've listed wrongly under the reflexive closure, since from $(0,8)$ and $(8,0)$ you've got to add $(0,0)$. From $(8,0)$ and $(0,8)$, you got to add $(8,8)$, and so on with the other numbers in the field.
Lastly, I thought you might like to see the figure I created illustrating the various kinds of closures of an example relation (not the same as your example). Some of my students found it useful to notice the differences in the various kinds of closures, and to discover the reason each diagram was as it was. I believe that anyone who can understand this picture really knows the reflexive, transitive and symmetric closures of a relation and their combinations.
I made a small post at https://plus.google.com/u/0/+JoelDavidHamkins1/posts/Gu3eScwxDzf.