Final answer:
Given the properties of the relation R, including identity and symmetric pairs, we can deduce that R² will contain all the pairs in R and is therefore at least a superset of R.
Step-by-step explanation:
The relation R on set A is defined as R = {(1,1), (2,2), (3,3), (4,4), (1,3), (3,1), (2,4), (4,2)}. To determine the nature of R² without explicitly calculating it, we recognize that R includes all of the identity pairs (x, x) for each element x in the set A, which is a characteristic of the identity relation. Furthermore, for each non-identity pair (x, y) in R, the inverse pair (y, x) is also in R, suggesting reflexivity and symmetry.
Considering these properties, R², which is the composition of R with itself, would contain at least all of the same pairs as R since composing any identity pair with itself yields the same pair and composing a pair and its inverse will yield the identity pairs. Therefore, without explicitly calculating R², we can deduce that R² is at least a superset of R, if not equal.
Thus, the correct answer is C. superset. R²≥R.