...I've also prepared students for challenging competitions like the Russian Regional Math Olympiads, AMC, and Math Counts. My journey in mathematics started with a degree in Mathematics, and I've taught in diverse locations worldwide. I have a deep passion for teaching and traveling, and I began my teaching career in Russia, working with gifted high school students. (read more)
...It will now be shown that if R on A is antisymmetric, R n R-1 ? {(a, a): a ? A}. Let m, n ? A. Suppose that R on A is antisymmetric. Let R = {(m, n): m, n ? A}. Therefore, by definition of inverse relation, R-1 = {(n, m): m, n ? A}. Let (m, n) ? (R n R-1). By definition of intersection, (m, n) ? R and (m, n) ? R-1. (read more)