Answer to Question #312603 in Discrete Mathematics for pempem

Question

Show, by the use of replacement rules, that (-p ^ q) ^ (q→p) = F

 are logically equivalent.

Answer 1

Solution

For the left side (¬p ∧ q) ∧ (q→p)

 

(q→p) = (¬p→¬q)                           Contraposition

= (¬(¬p) ∨¬q)                        material implication

= (p ∨¬q)                               ¬(¬p) = p

= ¬ (¬p ∧¬(¬q))                    Dem organs laws

= ¬ (¬p ∧q)                          ¬(¬q) = q

            = ¬ A                                     consider (¬p ∧q) = A

Therefore,

(¬p ∧ q) ∧ (q→p)     = A ∧¬ A

                                   = F                 (always false)

 

Hence

(¬p ∧ q) ∧ (q→p) = F

The same has been shown by the truth table below that (-p ∧ q) ∧ (q→p) = F