Back To Basics (3) Proof by Tautology

Tautologies can be used to prove new propositions or justify old ones. The transitive property of both ⇒ and ⇔ allow this to be done. Start with truth tables to demonstrate their transitivity, then use proof by tautology. (Tautologies listed on pages tab at the top right corner of the Home page)

Proof by Tautology 1

1.) A ∧ B                                         Assume

2.) ¬[¬(A ∧ B)]                             t8

3.) ¬[(¬A ∨ ¬B)]                           De Morgan's First Law

 ∴ (A ∧ B) ⇔ ¬[(¬A ∨ ¬B)]        Conclusion

Each line in this proof is equivalent to the next. The transitive property allows for the "detachment" and/or "addition" of propositions. Since each line from the first is a tautology built from line one, each proposition must therefore be equivalent to the first.

Proof by Tautology 2

1.) ¬A ∧ ¬B                             Assume

2.) ¬{[¬(¬A)] ∨ ¬(¬B)]}       Proof by Tautology 1

3.) ¬{A ∨ B}                            Complete Law of Double Negation

∴ ¬A ∧ ¬B ⇔ ¬{A ∨ B}        Conclusion