Antisymmetric relations

Let R and S be antisymmetric relations on a set A. Does R union S have to be antisymmetric also? Give a counterexample if the answer is no and proof if it is yes.

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...symmetry means that the only case where xRy ^ yRx can occur is if
x=y. So it allows for (but not required to have) xRx in the relation
but not both of xRz and ...