Let . Let be a relation on the set . Use the matrix representation of relations in order to compute the following:
a) Let be the reflexive closure of . Compute .
My current answer is
b) Let be the transitive closure of . Compute . (I used Warshall's algorithm)
My current answer is
c) Let . Compute the symmetric closure of and call this set .
My current answer is
d) Prove or disprove that is an equivalence relation on the set . If is an equivalence relation, then list the elements in each of the equivalence classes. If is not an equivalence relation, then list what elements need to be added to to make it an equivalence relation.
(I have not completed this part yet, so I could use a hand here)
I mainly need to know whether my work on the first three parts has yielded correct answers (it'd be pretty easy to make a mistake). As for the last part, any help would be appreciated.
Yeah, must be the smallest transitive relation that contains .
Using the method here (under 'Existence and description'), I was able to get
(might want to check it though).
Now to get , you just use the Boolean operation " " on each of the corresponding entries of and (since the 1,1 entry of is 1 and for it's 0, then the 1,1 entry of is: , and so on).
Finally, if is a symmetric matrix, then it is an equivalence relation (since is reflexive and transitive).
Ran through it and got a corrected answer. My new answer for (which is the symmetric closure of ) is as follows:
Now I just need to do the last part for it: prove or disprove that is an equivalence relation on the set and, if it is an equivalence relation, list the elements in each of its equivalence classes. Easier said than done, so any assistance would be appreciated.