Let p and q be propositions. (i) Show (p →q) ≡ (p ∧ ¬q) →F (ii.)...

Let p and q be propositions.

(i) Show (p →q) ≡ (p ∧ ¬q) →F

(ii.) Why does this equivalency allow us to use the proof by contradiction technique?

Solutions

Expert Solution

(i) p and q are two propositions.
Let us prove that p q and ( p ¬q F ) are equivalent using a Truth Table.

p q ¬q p ¬q F p ¬q F p q

0 0 1 0 0 1 1
0 1 0 0 0 1 1
1 0 1 1 0 0 0
1 1 0 0 0 1 1

Since the corresponding truth values of p q and p ¬q F are the same, hence (p q) ≡ (p ¬q) F

(ii) The technique of Proof by Contradiction is an Indirect Proof technique. To prove an implication P Q,
we assume (on the contrary) that P and NOT Q are both true. Under this antecedent, we prove that the Consequent is a Contradiction F (that is, a statement F which is always false, in some given logical system). Since a Contradiction arises, this therefore implies that P Q is indeed true. Hence, the equivalency P Q ( P ¬Q F ) allows us to use the proof by contradiction technique.

Colby Messinger answered 1 year ago

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