I need some help in regards to symmetric and transitive closures.
So lets suppose that we have a relation $R = \{(1,2), (2,1), (3,1)\}$ on a set $X = \{1,2,3,4\}$
I know that if we want to create the reflexive closure of the set, then we to include $(1,1), (2,2), (3,3)$ and $(4,4)$. So even though the number $4$ is missing from the original relation, we still need to include it in the reflexive closure. I have learned this from the wikipedia page for relfexive closures: https://en.wikipedia.org/wiki/Reflexive_closure
Now my question is the following: Does this hold true for symmetric and transitive closures as well? If an element is missing in the original relation, do we need to include it in its symmetric and transitive closures?
Best Answer
The answer is no.
Symmetric closure means that if $x\sim y$, then you add $y\sim x$ to the relation.
Similarly, transitive closure means that if both $x\sim y$ and $y\sim z$ are in the relation, then you add $y \sim z$.
By definition, you need them to be present in the relation already.