Truth Tables
Suppose you are given two logical expressions, e.g., $p \to q$ and $\neg q \to \neg p$, and want to check if they are equivalent or not. You can do so by constructing the truth tables for each of the 2 expressions:
| $p$ | $q$ | $\neg p$ | $\boldsymbol{\neg p \vee q}$ |
|---|---|---|---|
| $0$ | $0$ | $1$ | $1$ |
| $0$ | $1$ | $1$ | $1$ |
| $1$ | $0$ | $0$ | $0$ |
| $1$ | $1$ | $0$ | $1$ |
| $p$ | $q$ | $p'$ | $q'$ | $\boldsymbol{\neg \neg q \vee \neg p = q \vee \neg p}$ |
|---|---|---|---|---|
| $0$ | $0$ | $1$ | $1$ | $1$ |
| $0$ | $1$ | $1$ | $0$ | $1$ |
| $1$ | $0$ | $0$ | $1$ | $0$ |
| $1$ | $1$ | $0$ | $0$ | $1$ |
Because the sequences of truth values that we got under the $\neg p \vee q$ and $\neg \neg q \vee \neg p = q \vee \neg p$ columns are exactly the same: $\langle1, 1, 0, 1\rangle$, this means that the statement are equivalent, so $p \to q = \neg q \to \neg p$ is a valid equivalence!
The next 2 slides introduce additional such equivalence; we use $\Leftrightarrow$ to indicate that the two expressions are equivalent.
These notes by Miriam Briskman are licensed under CC BY-NC 4.0 and based on sources.