Final answer:
An asymmetric relation is always antisymmetric because if a is related to b, then b cannot be related to a. However, an antisymmetric relation is not necessarily asymmetric because it allows reflexive pairs where a equals b.
Step-by-step explanation:
An asymmetric relation is one in which, if a is related to b, then b cannot be related to a. In other words, if (a, b) is in the relation, then (b, a) is not. This means that there is no pair of distinct elements such that both are related to each other. Considering this definition, an asymmetric relation is inherently antisymmetric because antisymmetry requires that if (a, b) and (b, a) are in the relation, then a must equal b. However, since (b, a) can never be in an asymmetric relation if (a, b) is, the condition for antisymmetry is vacuously satisfied.
On the other hand, an antisymmetric relation does not have to be asymmetric. Antisymmetry allows for the possibility of (a, b) and (b, a) being in the relation as long as a equals b. This means reflexive pairs like (a, a) are allowed in an antisymmetric relation but are not allowed in an asymmetric relation because they violate the definition of asymmetry.
For example, in a set of numbers, the less than relation is both asymmetric and antisymmetric because if a < b, we cannot have b < a. However, the divides relation is antisymmetric but not asymmetric; for instance, 4 divides 12, but 12 does not divide 4, satisfying antisymmetry. However, since any number divides itself, it is not asymmetric.