Final answer:
The absence of symmetry in a relation does not imply that the relation is antisymmetric, and the absence of antisymmetry does not imply that the relation is symmetric, as illustrated by counterexamples for both implications.
Step-by-step explanation:
The question revolves around the properties of symmetry and antisymmetry in relations on sets. A relation R is considered symmetric if whenever (a, b) is in R, then (b, a) is also in R. On the other hand, R is antisymmetric if for any (a, b) and (b, a) in R, a must be equal to b. To address whether the absence of symmetry implies antisymmetry, and vice versa, we need to look at each implication separately.
A counterexample for the first statement, where R not being symmetric implies R being antisymmetric, is as follows:
- Suppose A = \{1, 2\} and R = \{(1, 2)\}. Here R is not symmetric because (2, 1) is not in R, but it's also not antisymmetric because there's no pair (b, a) to compare with (a, b). This shows that the lack of symmetry does not guarantee antisymmetry.
For the reverse implication, consider A = {1, 2, 3} and a relation R = \{(1, 2), (2, 1), (2, 2)\}. This relation is not antisymmetric because (1, 2) and (2, 1) are in R but 1 is not equal to 2. It is also not symmetric because, for example, (2, 3) is not in R while (3, 2) is not even an element to consider. This shows that lack of antisymmetry doesn’t imply symmetry.
Therefore, not being symmetric doesn't necessarily make a relation antisymmetric, and not being antisymmetric doesn't necessarily make a relation symmetric.