Good day. Just a couple of questions on Equivalence Relations/Classes and proofs.
Let X denote the number of ways to arrange 4 distinct elements (this was 24 as obtained via 4!)
Consider the relation R between arrangements such that a new arrangement can be obtained by shifting all elements one space to the right. (1234 => 4123)
If A on X is the transitive closure of relation R
- Would it be correct to say that A denotes "The number of possible arrangements obtained by shifting all distinct elements to the right"?
I'm not sure if the idea of "24 different arrangements" is necessary as A is a transitive closure.
- Suppose A is an equivalence relation. How many distinct equivalence classes exist for A? Which classes are they?
This is something I don't really understand. Are the permutations of the 4 elements the same as the "equivalence classes"? Since shifting all elements to the right doesn't affect the permutations prima facie.
- Prove A is an equivalence relation.
Reflexive - ???
Symmetric - "There are 24 ways to arrange the distinct elements after shifting them to the right" and "There are 24 ways to arrange the elements after shifting them to the left" Hence, they are symmetrical.
Transitive - "There are 24 ways to arrange the distinct elements" and "All elements can be shifted to the right to create a new arrangement". This is transitive as A is the transitive closure of these two statements.
Any help/comments would be greatly appreciated.