Equivalences
Let's check if the expressions: $p \vee q \wedge r$ and $q \vee p \wedge r$ are equivalent or on:
| $p$ | $q$ | $r$ | $q \wedge r$ | $\boldsymbol{p \vee q \wedge r}$ |
|---|---|---|---|---|
| $0$ | $0$ | $0$ | $0$ | $0$ |
| $0$ | $0$ | $1$ | $0$ | $0$ |
| $0$ | $1$ | $0$ | $0$ | $0$ |
| $0$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $0$ | $0$ | $0$ | $1$ |
| $1$ | $0$ | $1$ | $0$ | $1$ |
| $1$ | $1$ | $0$ | $0$ | $1$ |
| $1$ | $1$ | $1$ | $1$ | $1$ |
| $p$ | $q$ | $r$ | $p \wedge r$ | $\boldsymbol{q \vee p \wedge r}$ |
|---|---|---|---|---|
| $0$ | $0$ | $0$ | $0$ | $0$ |
| $0$ | $0$ | $1$ | $0$ | $0$ |
| $0$ | $1$ | $0$ | $0$ | $1$ |
| $0$ | $1$ | $1$ | $0$ | $1$ |
| $1$ | $0$ | $0$ | $0$ | $0$ |
| $1$ | $0$ | $1$ | $1$ | $1$ |
| $1$ | $1$ | $0$ | $0$ | $1$ |
| $1$ | $1$ | $1$ | $1$ | $1$ |
Because the sequence of digits that we got under the $p \vee q \wedge r$ column is different from the one under the $q \vee p \wedge r$ column, these two expressions aren't equivalent. We say: $p \vee q \wedge r \not\Leftrightarrow q \vee p \wedge r$ (
$\not\Leftrightarrow$
These notes by Miriam Briskman are licensed under CC BY-NC 4.0 and based on sources.