Proving $R^n$ is antisymmetric when R is antisymmetric
Solution 1:
$S=\{1,2,3,4\}$
$R = \{(1,3), (3,2), (2,4), (4,1)\}$
$R^2 = \{(1,2), (3,4), (2,1), (4,3)\}$
$R$ is antisymmetric, however $R^2$ is not antisymmetric, therefore disproven by counterexample.
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Solution 2:
HINT: Construct a counterexample for $n=2$; you can take $S$ to be a $4$-element set and $R$ to contain exactly $4$ ordered pairs.