The statement ( p ∧ ( p → q ) ∧ ( q → r ) ) → r is

The statement

( p ∧ ( p \to q ) ∧ ( q \to r ) ) \to r

Options

Aa tautology
Bequivalent to $q \to ~ r$
Ca fallacy
Dequivalent to $p \to ~ r$

Correct answer

A. a tautology

Step-by-step solution

Let $f = ( p ∧ ( p \to q ) ∧ ( q \to r ) )$

Make truth table,

$\begin{cases}r & p & q & p \to q & q \to r & f & f \to r \\ T & T & T & T & T & T & T \\ F & T & T & T & F & F & T \\ T & T & F & F & T & F & T \\ F & T & F & F & T & F & T \\ T & F & T & T & T & F & T \\ F & F & T & T & F & F & T \\ T & F & F & F & T & F & T \\ F & F & F & F & T & F & T\end{cases}$

So, the answer is tautology.

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