Recitation 3

In addition to answering questions from students in section C and D, there were couple of other things going on.

In section C, I introduced a more general version of AGM inequality. Suppose x_1, x_2, \ldots, x_n > 0, then \sqrt[n]{x_1\cdot\ldots\cdot x_n}\leq \frac{x_1+\ldots+x_n}{n}.

And we proved the case for n=4 by using the original AGM inequality. In fact, this technique can be used to prove the cases for n=2^m where m is a natural number. But yet we haven’t proved for the case when n=3.

In section D, I explored a practical method to calculate sets. The basic rules behind the scenes are:

1. Associativity: (A \cup B)\cup C = A\cup (B \cup C) and (A \cap B)\cap C = (A \cap B)\cap C.
2. Commutativity: A \cup B = B \cup A and A \cap B = B \cap A.
3. Distributivity: A\cap(B\cup C) = (A\cap B)\cup(A\cap C) and A\cup(B\cap C) = (A\cup B)\cap(A\cup C).
4. Identity: A\cup\emptyset = A.
5. Annihilator: A\cap\emptyset = \emptyset.
6. Idempotence: A\cup A = A\cap A = A.
7. Absorption: A\cap(A\cup B)=A and A\cup(A\cap B)=A.
8. Complementation: A\cap \bar{A}=\emptyset.
9. Double negation: \bar{\bar{A}}=A.
10. De Morgan: \bar{A}\cap\bar{B}=\overline{A\cup B} and \bar{A}\cup\bar{B}=\overline{A\cap B}.