Question

Using the laws of logic to prove tautologies.Use the laws of propositional logic to prove that each statement is a tautology.a. ¬r ∨ (¬r → p)b. ¬(p → q) → ¬q

Answer 1

Answer: The proofs are given below.

Step-by-step explanation: We are given to prove that the following statements are tautologies using truth table :

(a) ¬r ∨ (¬r → p) b. ¬(p → q) → ¬q

We know that a statement is a TAUTOLOGY is its value is always TRUE.

(a) The truth table is as follows :

r p ¬r ¬r→p ¬r ∨ (¬r → p)

T T F T T

T F F T T

F T T T T

F F T F T

So, the statement (a) is a tautology.

(b) The truth table is as follows :

p q ¬q p→q ¬(p→q) ¬(p→q)→q

T T F T F T

T F T F T T

F T F T F T

F F T T F T

So, the statement (B) is a tautology.

Hence proved.