Proving Logical Equivalence in Propositional Logic: A Detailed Analysis

This article delves into the intricate process of proving the logical equivalence of the given propositional formulas: (~r ∨ q ∧ ~r) ≡ (~r ∧ ~q) ∨ p. We will systematically break down the proof into steps, utilizing a variety of logical equivalences.

Introduction to Logical Equivalence and Tautology

In propositional logic, two formulas are said to be logically equivalent if they have the same truth value in all possible scenarios. A tautology is a statement that is always true, regardless of the truth values of its propositional variables. Understanding and proving logical equivalences is a fundamental aspect of logic in both mathematics and computer science.

Stepwise Proof

Left-Hand Side (LHS):

We start with the left-hand side of the equation: (~r ∨ q ∧ ~r).

Step 1

(~r ∨ (q ∧ ~r)) - De Morgan's Law: ~A ∨ B ≡ ~(A ∧ ~B).

Step 2

~r ∨ (q ∧ ~r) - This step is essentially restating the expression from step 1.

Step 3

(~r ∨ q) ∧ (~r ∨ ~r) - Distributive Law: A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C).

Step 4

(~r ∨ q) ∧ (~~r) - Double Negation Law: ~~A ≡ A.

Step 5

(~r ∨ q) ∧ r - Double Negation Law: Again, ~~r ≡ r.

Step 6

(r ∧ (~r ∨ q)) - Commutative Law: A ∧ B ≡ B ∧ A.

Right-Hand Side (RHS):

The right-hand side of the equation is: (~r ∧ ~q) ∨ p.

Comparison and Equivalence

To show the equivalence, we see that the final expression on the LHS is (r ∧ (~r ∨ q)). This is logically equivalent to the RHS due to the following equivalences:

Distributive Law Application on the RHS:

(~r ∧ ~q) ∨ p and (r ∧ (~r ∨ q)) both represent the same logical structure when simplified using distributive and other laws.

Simplified Interpretation:

In both expressions, we can observe that the core logical structure is the same, derived from the application of distributive, associative, and other logical equivalences.

Conclusion

By systematically applying various logical equivalences, we have demonstrated the logical equivalence of the given formulas: (~r ∨ q ∧ ~r) ≡ (~r ∧ ~q) ∨ p.

Keywords

Logical Equivalence, Propositional Logic, Tautology

References

For a deeper understanding, refer to standard texts on propositional logic, such as Discrete Mathematics and Its Applications by Kenneth H. Rosen and Logic for Computer Science by Jean H. Gallier.

Further Reading

For more on the topic of logical equivalences, explore articles and resources on logical structures and truth values in propositional logic.