Question

The proposition of \[p \Rightarrow \sim \left( {{p^{ \sim q}}} \right)\] is
A. Contradiction
B. A tautology
C. Either (A) or (B)
D. Neither (A) or (B)

Answer 1

Hint: In this question, we make a truth table for the given proposition and see if all of the items are true, all of them are false, or none of them are true.

Complete step-by-step solution:
We know that tautology is defined as a formula that is "always true," that is, it is true for any assignment of truth values to its simple components. A tautology can be thought of as a logical rule. A tautology's inverse is a contradiction, a formula that is "always false." Contingency refers to a proposition that is neither a tautology nor a contradiction.
We need to find the proposition of \[p \Rightarrow \sim \left( {{p^{ \sim q}}} \right)\]
Now we have to solve this proposition by constructing the truth table

\[p\]	\[q\]	\[ \sim q\]	\[{p^{ \sim q}}\]	\[ \sim \left( {{p^{ \sim q}}} \right)\]	\[p \Rightarrow \sim \left( {{p^{ \sim q}}} \right)\]
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 \[p \Rightarrow \sim \left( {{p^{ \sim q}}} \right)\] is neither tautology nor contradiction.
Hence, option (D) is correct.

Additional information: A truth table is a table used in logic, specifically Boolean algebra, that lists the functional values for each of their functional arguments.

Note: Students must read about what is a tautology, contradiction, and contingency meaning. To master these types of difficulties, practice more problems. It's important to remember that if a composite proposition is a contingent, it can't be a tautology or a contradiction.