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...Suppose R n R-1 ? {(a, a): a ? A}. Let R = {(x, y): x, y ? A}. By definition of inverse relation, R-1 = {(y, x): x, y ? A}. By definition of subset, if (x, y) ? R n R-1, (x, y) ? {(a, a): a ? A}. Hence, x = y, and so (x, y) ? R implies (y, x) ? R. Since x = y, ? (xRy ? yRx), x = y. Thus, by definition of antisymmetric, R on A is antisymmetric. (read more)