Final answer:
1. R is anti-symmetric: False. 2. R is an equivalence relation: False. 3. R is symmetrical: True. 4. R is reflexive: True. 5. R is transitive: True. 6. R is a partial order: False.
Step-by-step explanation:
1. R is anti-symmetric: False. To determine if a relation is anti-symmetric, we need to check if for every pair (a, b) in the relation, if aRb and bRa, then a = b. In this case, (9, 2) and (2, 9) are in the relation, but 9 is not equal to 2, so R is not anti-symmetric.
2. R is an equivalence relation: False. To be an equivalence relation, R must be reflexive, symmetric, and transitive. While R is reflexive and symmetric, as (a, a) and (a, b) implies (b, a) are in R, it is not transitive as (9, 2) and (2, 9) are in R, but (9, 9) is not in R, violating the transitive property.
3. R is symmetrical: True. Symmetric means that for every pair (a, b) in the relation, if aRb, then bRa. In this case, (9, 2) implies (2, 9), and (2, 9) implies (9, 2), so R is symmetrical.
4. R is reflexive: True. Reflexive means that for every element a in A, (a, a) is in R. In this case, (0, 0), (2, 2), (6, 6), and (9, 9) are all in R, so R is reflexive.
5. R is transitive: True. Transitive means that if (a, b) and (b, c) are in R, then (a, c) is in R. In this case, (2, 9) and (9, 2) are in R, and this implies (2, 2) is in R, so R is transitive.
6. R is a partial order: False. To be a partial order, R must be reflexive, antisymmetric, and transitive. We have already determined that R is reflexive and transitive, but since it is not antisymmetric, it cannot be a partial order.