Final answer:
The relation n4 is reflexive, not symmetric, not antisymmetric, and not transitive. The same properties apply to n8.
Step-by-step explanation:
When determining if a relation is reflexive, symmetric, antisymmetric, or transitive, we need to consider the properties of the relation and the elements involved. In this case, we are dealing with the relation n4.
A relation is reflexive if every element is related to itself. Since n4 is simply a stand-in for a number, let's consider an example with a specific number. For example, let's assume n4 represents the number 4. In this case, 4 is related to itself because 4 to the power of 4 is equal to 4^4 = 256, which is indeed the number 4. Therefore, the relation n4 is reflexive.
The relation is symmetric if for every pair (a, b) in the relation, the pair (b, a) is also in the relation. Considering our example with n4 as 4, we can see that 4^4 = 256 and 256^4 = 18446744065119617000. While both 4^4 and 256^4 are valid, they are not equal. Therefore, the relation n4 is not symmetric.
A relation is antisymmetric if for every pair (a, b) in the relation, where a is not equal to b, if (b, a) is also in the relation, then a must be equal to b. Since the relation n4 is not symmetric, we can conclude that it is also not antisymmetric.
A relation is transitive if for every triple (a, b, c) in the relation, if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. Again, considering our example with n4 as 4, let's assume a = 4, b = 4, and c = 256. We can see that 4^4 = 256 and 256^4 = 18446744065119617000. While both 4^4 and 256^4 are valid, they are not equal. Therefore, the relation n4 is not transitive.
Given that n8 has the same properties as n4, it would also be reflexive, not symmetric, not antisymmetric, and not transitive.