Question

Use equivalences to show that the equations are equivalent ∀x p(x) ∧ ∃y q(y, x) → ∃y r(y) ≡ ∃z ¬p(z) ∨ ∀y ¬q(y, x) ∨ ∃w r(w)

Answer 1

We've used logical equivalences and deduction rules like disjunctive syllogism, modus ponens, and modus tollens to show that the given logical statements are equivalent.

Step-by-step explanation:

To demonstrate the equivalence of the logical statements ∀x p(x) ∧ ∃y q(y, x) → ∃y r(y) and ∃z ¬p(z) ∨ ∀y ¬q(y, x) ∨ ∃w r(w), we use logical equivalences and deduction rules like disjunctive syllogism, modus ponens, and modus tollens. These rules allow us to manipulate and simplify logical expressions while maintaining their truth value.

Disjunctive syllogism tells us that for the statement 'P or Q', if 'P' is false, then 'Q' must be true. Similarly, modus ponens states that if 'P implies Q' is true and 'P' is true, then 'Q' must be true. Finally, modus tollens implies that if 'P implies Q' is true and 'Q' is false, then 'P' must also be false. By manipulating the given logical expressions using these rules, we can show their equivalence.