[[Logical Operations]] Two expressions are equivalent if for every possible way to input truth values into the variables, the statements have the same truth value.
We can check this with a truth table.
Claim: ~(P$\land$Q)$\equiv$~P$\land$~Q
| P | Q | ~P | ~Q | P$\land$Q | ~(P$\land$Q) | ~P$\land$~Q |
|---|---|---|---|---|---|---|
| T | T | F | F | F | F | F |
| T | F | F | T | F | T | F |
| F | T | T | F | F | T | F |
| F | F | T | T | F | T | T |
| This claim is false. |
If the truth tables differ between any two logical statements, they are not equivalent.
Claim: ~(P$\land$Q)$\equiv$P$\oplus$Q
| P | Q | P$\land$Q | ~(P$\land$Q) | P$\oplus$Q |
|---|---|---|---|---|
| T | T | T | F | F |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | T | F |
| This claim is false. |
Claim: ~(P$\land$Q)$\equiv$~P$\lor$~Q
| P | Q | ~P | ~Q | ~(P$\land$Q) | ~P$\lor$~Q |
|---|---|---|---|---|---|
| T | T | F | F | F | F |
| T | F | F | T | T | T |
| F | T | T | F | T | T |
| F | F | T | T | T | T |
| This claim is true. |
Since the two expressions yield the same output for any input, they are logically equivalent. [[Logical Equivalence]]