Formal Proof
The laws may be proven directly using truth tables; "1" represents true, "0" represents false.
First we prove: ¬(p ∨ q) ⇔ (¬p) ∧ (¬q).
| p | q | p ∨ q | ¬(p ∨ q) | ¬p | ¬q | (¬p) ∧ (¬q) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
Since the values in the 4th and last columns are the same for all rows (which cover all possible truth value assignments to the variables), we can conclude that the two expressions are logically equivalent.
Now we prove ¬(p ∧ q) ⇔ (¬p) ∨ (¬q) by the same method:
| p | q | p ∧ q | ¬(p ∧ q) | ¬p | ¬q | (¬p) ∨ (¬q) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
Read more about this topic: De Morgan's Laws
Famous quotes containing the words formal and/or proof:
“Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more formal one said, “Do you think the Holy Dove could fly down with only one wing?””
—Horace Walpole (1717–1797)
“Right and proof are two crutches for everything bent and crooked that limps along.”
—Franz Grillparzer (1791–1872)