I have a particular question in regard to my preparation for an upcoming exam. Unfortunatelly I have no one to ask or check/compare my answers, so here it is:
On the set $U = \{1, 2, 3, 4, 5, 6\}$ define the relation
$$R = \{(i, i) \;|\; i ∈ U\} ∪ \{(1, 2),(2, 1),(2, 3),(3, 2),(1, 3),(3, 1),(4, 6),(6, 4)\}$$
Show that R is an equivalence relation. Establish what are the equivalence
classes of R, in particular, how many equivalence classes R has, and how many
elements each of them has.
So $R = \{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1, 2),(2, 1),(2, 3),(3, 2),(1, 3),(3, 1),(4, 6),(6, 4)\}$
An equivalence relation is
(1) reflexive – which R is
(2) symmetric
(3) transitive.
Here is where I got confused, because I think that R is not an equivalence relation, because for it to be symmetric (definiton of symmetric is $\forall x,y\in U \;:\; (x,y)\in R \Longleftrightarrow (y,x)\in R$) but there is no pair including 5, and 5 is in U, hence its not symmetric. Or should I only work with the tuples given and not consider every element in U? I hope I explained my concerns correctly.
Also, if it is the latter, and R is indeed an equivalence relation, then are these the corect equivalence classes?
$[1]=[2]=[3]= \{1,2,3\}$ and $[4] = [6] = \{4,6\}$ .
Thank you in advance.
Best Answer
For symmetry it is enough to ask by each tuple $(a,b)$ in $R$ the question: "is $(b,a)$ also in $R$?"
If the answer is "yes" for all tuples in $R$ then $R$ is symmetric.
You forgot equivalence class $[5]=\{5\}$.
Further it is okay.