Question

identify the correct steps involved while proving (p ∨ ¬q) ∧ (q ∨ ¬r) ∧ (r ∨ ¬p) is true when p, q, and r have the same truth value and it is false otherwise.

Answer 1

To prove whether the expression (p ∨ ¬q) ∧ (q ∨ ¬r) ∧ (r ∨ ¬p) is true or false, we need to consider two cases: when p, q, and r are all true, and when they are all false.

Step-by-step explanation:

To prove that the expression (p ∨ ¬q) ∧ (q ∨ ¬r) ∧ (r ∨ ¬p) is true when p, q, and r have the same truth value and false otherwise, we need to consider two cases:

1. If p, q, and r are all true.

   • In this case, the expression evaluates to (true ∨ ¬true) ∧ (true ∨ ¬true) ∧ (true ∨ ¬true).

   • Simplifying further, we get true ∧ true ∧ true, which is true.

2. If p, q, and r are all false.

   • In this case, the expression evaluates to (false ∨ ¬false) ∧ (false ∨ ¬false) ∧ (false ∨ ¬false).

   • Simplifying further, we get false ∧ false ∧ false, which is false.

Therefore, the expression (p ∨ ¬q) ∧ (q ∨ ¬r) ∧ (r ∨ ¬p) is true when p, q, and r have the same truth value and false otherwise.