De Morgan's Laws
Continuing with the relationship between ∧ and ∨, we can also use De Morgan's Law to rearrange sentences involving negation in conjunction with these connectives. Actually, there are 2 equivalences which taken as a pair are called De Morgan's Law:
¬(P ∧Q) ≡ ¬P ∨¬Q
¬ (P ∨ Q) ≡ ¬P ∧ ¬Q
These are essential rules and it is good spending some time thinking regarding why they are true.
The contraposition equivalence is following:
P -> Q ≡ ¬Q -> ¬P
At first, this can seem a little strange because it seem that we have said nothing in the first sentence regarding ¬Q, soin the second sentence how can we infer anything from it? However, imagine we know that P implies Q, and we saying that Q was false. In this case, if we were to imply that P was true because we know that P implies Q, we also know that Q is true. But Q was false .So we cannot possibly denote that P is true, which means that we ought to imply that P is false (because we are in propositional logic, so P might be either true or false). This argument shows that we may replace the first sentence by the second one, and it is left as aworkout to construct a similar argument for the vice-versa part of this equivalence.