Question

Show that each of these conditional statements is a tautology

by using truth tables.

a) [¬p ∧ (p ∨ q)] → q

b) [(p → q) ∧ (q → r)] → (p → r)

c) [p ∧ (p → q)] → q

d) [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r

Answer 1

Solution (a) The truth table of $[¬p ∧ (p ∨ q)] → q$ is shown below

The truth table shows that $[¬p ∧ (p ∨ q)] → q$ is a tautology

(b) The truth table of $[(p → q) ∧ (q → r)] → (p → r)$ is shown below

The truth table shows that $[(p → q) ∧ (q → r)] → (p → r)$ is a tautology

(c) The truth table of $[p ∧ (p → q)] → q$ is shown below

The truth table shows that $[p ∧ (p → q)] → q$ is a tautology

(d) The truth table of $[(p ∨ q) ∧ (p → r) ∧ (q → r)] → r$ is shown below

The truth table shows that $[(p ∨ q) ∧ (p → r) ∧ (q → r)] → r$ is a tautology