Question

Choose the correct option.

Statement $1:(p\wedge \sim q)\wedge (\sim p\wedge q)$ is a fallacy.

Statement $2:(p\Rightarrow q)\iff (\sim q\Rightarrow \sim p)$ is a tautology.

• A. Both the statements are TRUE and STATEMENT $2$ is the correct explanation of STATEMENT $1.$
• B. Both the statements are TRUE but STATEMENT $2$ is NOT the correct explanation of STATEMENT $1.$
• C. STATEMENT $1$ is TRUE and STATEMENT $2$ is FALSE.
• D. STATEMENT $1$ is FALSE and STATEMENT $2$ is TRUE.

Answer 1

If a logical compound statement always produces the truth (true value), then it is called a tautology.

The opposite of tautology is called fallacy or contradiction, in which the compound statement is always false.
Statement $1:$

$(p\wedge \sim q)\wedge (\sim p\wedge q)$
Using Set Theory approach, this can be expressed as:
$(P\cap Q^c)\cap (P^c\cap Q)$



Clearly, the resultant is $\varphi$.
Hence, statement $1$ is a fallacy.

Statement $2:$

$(p\Rightarrow q)\iff (\sim q\Rightarrow \sim p)$
Clearly, $(\sim q\Rightarrow \sim p)$ is a contra-positive of $(p\Rightarrow q)$ and vice-versa.
Hence, statement $2$ is a tautology.

Hence, the correct option is C Both the statements are TRUE but STATEMENT $2$ is NOT the correct explanation of STATEMENT $1.$