Question

Do we necessarily get an equivalence relation when we form the transitive closure of the symmetric closure of the reflexive closure of a relation

Answer 1

Answer:

Explanation:Transitivity says that if there are elements a,b,c (any of which can be equal) such that ⟨a,b⟩∈R and ⟨b,c⟩∈R, then ⟨a,c⟩∈R. This condition is vacuous – says nothing – if there are no elements a,b, and c such that ⟨a,b⟩∈R and ⟨b,c⟩∈R. That’s the case here: the hypothesis of the if-then is never satisfied, so the then part is never invoked. To put it differently, if R were not transitive, there would have to be a,b, and c such that ⟨a,b⟩∈R and ... ⟨b,c⟩∈R, but ⟨a,c⟩∉R. And there certainly aren’t, since there aren’t even a,b, and c such that ⟨a,b⟩∈R and ⟨b,c⟩∈R.