Question

What is the statement \[\left( {p \to \left( {q \to p} \right)} \right) \to \left( {p \to \left( {p \vee q} \right)} \right)\] equivalent to?
A.\[\left( {p \vee q} \right) \wedge \left( { \sim p} \right)\]
B.\[\left( {p \vee q} \right) \vee \left( { \sim p} \right)\]
C. a contradiction
D. A tautology

Answer 1

Hint: To solve the question we will use the truth table, and truth table is a table which tells that the statement containing a combination of statements is true or false based on the elemental statements it is made of. We will represent true as T and false as F.
Formula Used:
We will use the truth table conditions, i.e,
\[p \vee q\] is true if either\[p\] is true or\[q\] is true, and it is only false if both \[p\] and \[p\] are false.
\[q \to p\] is true when both \[p\] is true or \[q\] is true, when\[q\] is false or \[p\] is true, and when both \[p\] and \[q\] are false.
Complete step by step solution:
Given statement is \[\left( {p \to \left( {q \to p} \right)} \right) \to \left( {p \to \left( {p \vee q} \right)} \right)\],
First let us take the possibilities for the two statements here \[p\],\[q\].Both can be true and false both. So, we can have 4 possibilities.
We will create a table for the possibilities

\[p\]	\[q\]
T	T
T	F
F	T
F	F

Now we will find the possibilities for \[q \to p\]i.e.,

\[p\]	\[q\]	\[q \to p\]
T	T	T
T	F	T
F	T	F
F	F	T

Now we find the possibilities for \[p \to \left( {q \to p} \right)\]i.e.,

\[p\]	\[q\]	\[q \to p\]	\[p \to \left( {q \to p} \right)\]
T	T	T	T
T	F	T	T
F	T	F	T
F	F	T	T

Now we will make the possibilities for\[p \vee q\], i.e.,

\[p\]	\[q\]	\[q \to p\]	\[p \to \left( {q \to p} \right)\]	\[p \vee q\]
T	T	T	T	T
T	F	T	T	T
F	T	F	T	T
F	F	T	T	F

Now we will make the possibilities for \[p \to \left( {p \vee q} \right)\]i.e,

\[p\]	\[q\]	\[q \to p\]	\[p \to \left( {q \to p} \right)\]	\[p \vee q\]	\[p \to \left( {p \vee q} \right)\]
T	T	T	T	T	T
T	F	T	T	T	T
F	T	F	T	T	T
F	F	T	T	F	T

Now we will make the possibilities for \[\left( {p \to \left( {q \to p} \right)} \right) \to \left( {p \to \left( {p \vee q} \right)} \right)\], i.e.,

\[p\]	\[q\]	\[q \to p\]	\[p \to \left( {q \to p} \right)\]	\[p \vee q\]	\[p \to \left( {p \vee q} \right)\]	\[\left( {p \to \left( {q \to p} \right)} \right) \to \left( {p \to \left( {p \vee q} \right)} \right)\]
T	T	T	T	T	T	T
T	F	T	T	T	T	T
F	T	F	T	T	T	T
F	F	T	T	F	T	T

So, from the table all possibilities are true so, the given condition is a tautology.
The correct option is D
Note: Students are often confused in Tautology and Contradiction.
1. Tautology: The truth table in which all the propositions are true in each row of the table is called tautology.
2. Contradiction: The truth table in which all the propositions are false in each row of the table is called contradiction.