Equivalences

We call the number $\boldsymbol 1$ and each one of the equivalences we've seen a tautology (= a statement that is always true). On the other hand, the false statement $\boldsymbol 0$ and any other false statement like $p \Leftrightarrow \neg p$ is called a contradiction.

Fun in-class exercises:

1. Prove the following equivalences by building truth tables for them:
   1. $p\wedge(q\vee r)\Leftrightarrow p\wedge q\vee p\wedge r$
   2. $p\vee p \wedge q\Leftrightarrow p$
2. Check if the following pairs of expressions are equivalent or not by building truth tables for them:
   1. $\neg (p \vee q)$ and $\neg p \vee \neg q$
   2. $p \wedge \neg p \vee \neg q$ and $p \wedge \neg q \vee \neg q$