# 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 $\bar{A}\cup\bar{B}=\overline{A\cap B}$.