...Since by definition of antisymmetric, (mRn ? nRm) implies m = n, then m = n. Notice that m and n are equal elements of A, and that (m, n) ? (R n R-1). Therefore, ? (m, n) ? (R n R-1), (m, n) ? {(a, a): a ? A}. Thus, by definition of subset, (R n R-1) ? {(a, a): a ? A}. Therefore, if R on A is... (read more)
...I was their primary caregiver throughout their childhood, and helped them with their schoolwork whenever they needed it. They both took math up through AP calculus (one of them took Calculus BC). One of them took AP Chemistry, the other took AP Physics, so I have seen some of the work done in those classes. My PhD is in Chemistry, 1988. (read more)
...In short, I try to teach mathematics in a way that helps students use it in their daily lives and see its relevance. While teaching at Syracuse University, I received the Outstanding Teaching award from the Graduate school in Spring 2022 for my teaching skills. I have taught multiple mathematic... (read more)