The property that for all numbers we have is called distributivity of over . Similar rules hold for and .
Let , , and be WFFs. Then
, and
.
Here are the truth tables for the four WFFs:
T | T | T | T | T |
T | T | F | T | T |
T | F | T | T | T |
T | F | F | F | F |
F | T | T | F | F |
F | T | F | F | F |
F | F | T | F | F |
F | F | F | F | F |
T | T | T | T | T |
T | T | F | T | T |
T | F | T | T | T |
T | F | F | T | T |
F | T | T | T | T |
F | T | F | F | F |
F | F | T | F | F |
F | F | F | F | F |
The last two columns are the same in both tables, so the formulas are logically equivalent. ∎
Let be a WFF. Then .
Let be a truth assignment for the propositional variables involved in . If then and so . Similarly if is false so is . Therefore under any truth assignment we have . ∎
Let and be WFFs. Then
, and
.
You might find it clearer to write the right hand sides of these equivalences as and , even though these are not well-formed formulas. From now on we will add or remove brackets from formulas where it helps to make them clearer or more readable even if it means that they are not strictly WFFs.
Again, proving this is simply a matter of checking the possibilities for the truth values of and under any assignment. In a table:
T | T | F | F |
---|---|---|---|
T | F | F | F |
F | T | F | F |
F | F | T | T |
The final columns are the same, so the two formulas have the same truth value no matter what truth assignment is used and are therefore logically equivalent. ∎
De Morgan’s laws can be generalized to more than two WFFs.
For any and any WFFs we have
, and
.