Question

The proposition $p \to \neg\left( {p \wedge \neg q} \right)$ is
A) Contradiction
B) Tautology
C) Either (A) or (B)
D) Neither (A) nor (B)

Answer 1

The given question can be solved by making use of a truth table.
To make a truth table draw a table with columns $p,q,\neg q,p \wedge \neg q,\neg\left( {p \wedge \neg q} \right),p \to \neg\left( {p \wedge \neg q} \right)$.
Now check whether the last column i.e. $p \to \neg\left( {p \wedge \neg q} \right)$ is a tautology or a contradiction.

Complete step by step solution:
It is asked to find the proposition $p \to \neg\left( {p \wedge \neg q} \right)$ .
So, we have to solve it by constructing the truth table for the above proposition.

p	q	$\neg q$	$p \wedge \neg q$	$\neg\left( {p \wedge \neg q} \right)$	$p \to \neg\left( {p \wedge \neg q} \right)$
T	T	F	F	T	T
T	F	T	T	F	F
F	T	F	F	T	T
F	F	T	F	T	T

Thus, the given proposition $p \to \neg\left( {p \wedge \neg q} \right)$ is neither a tautology nor a contradiction.

So, option (D) is the correct answer.

Note:
A truth table is a table used in logic, i.e. Boolean algebra, which sets out the functional values on each of their functional arguments. In general, a truth table is used to show whether an expression is true for all logical inputs.
A truth table has one column for each input variable and one final column showing all of the possible results of the logical operation that the table represents.