is not reflexive because not all the elements from A are related to one another in R
Correct. Reflexivity requires all elements in A to be self related. $\forall x\in A:(x,x)\in R$.
is symmetric because 2,2 can relate to itself
Incorrect. While $(2,2)$ is its own symmetrical pairing, Symmetry requires that all pairs that are in the relation have a corresponding symmetrical pairing. $\forall x\forall y: ((x,y)\in R \to (y,x)\in R)$
This is not so. $(2,7)$ is in the relation, but $(7,2)$ is not.
is anti-symmetric because similarly, 2,2 is symmetric can give 2=2
To be precise, anti-symmetry requires that all pairs in R have a cooresponding symmetrical pairing only if they are self-relations, such as for example $(2,2)$. $\forall x\forall y~(((x,y)\in R\land (y,x)\in R)\to x=y)$
is transitive because 2,2 can be paired with itself (2,2) (2,2) ⇒ (2,2)
Again it is for all cases. You cannot say it is so because one particular case holds. $\forall x\forall y\forall z~((x,y)\in R\land (y,z)\in R\to(x,z)\in R)$
So, therefore you must exhaustively check all possible cases to ensure no counter example exists to satisfy the negation of that; which is $\exists x\exists y\exists z~((x,y)\in R\land (y,z)\in R\land (x,z)\notin R)$
PS: no counter example exists.
Is my understanding correct?
Indications are no.
To ensure a universal statement is satisfied you must ascertain that all cases hold, rather than a single example.
On the other hand, finding a single counter example is all that is needed to prove a universal statement is not satisfied.