Match the following : \(\begin{array}{clcl} & \textbf{List-I} &…
2015
Match the following :
\(\begin{array}{clcl} & \textbf{List-I} & &\textbf{List-II} \\ \text{(a)} & \text{Vacuous} & \text{(i)} & \text{A proof that the implication $p \rightarrow q$ is true based} \\ &\text{proof}&&\text{on the fact that $p$ is false.} \\ \text{(b)} & \text{Trivial} & \text{(ii)} & \text{A proof that the implication $p \rightarrow q$ is true based} \\ &\text{proof}&&\text{on the fact that $q$ is true.} \\ \text{(c)} & \text{Direct} & \text{(iii)} & \text{A proof that the implication $p \rightarrow q$ is true that proceeds} \\ &\text{proof}&&\text{by showing that $q$ must be true when $p$ is true.} \\ \text{(d)} & \text{Indirect} & \text{(iv)} & \text{A proof that the implication $p \rightarrow q$ is true that proceeds} \\ &\text{proof}&&\text{by showing that $p$ must be false when $q$ is false.} \\ \end{array}\)
Codes :
- A.
(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)
- B.
(a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)
- C.
(a)-(iii), (b)-(ii), (c)-(iv), (d)-(i)
- D.
(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
Attempted by 59 students.
Show answer & explanation
Correct answer: A
Correct matching and explanation:
Vacuous proof: A proof that the implication p → q is true based on the fact that p is false. If the premise p never holds, the implication is true regardless of q.
Trivial proof: A proof that the implication p → q is true based on the fact that q is true. If the conclusion q always holds, the implication holds for every p.
Direct proof: A proof that proceeds by assuming p is true and showing that q must then be true. This directly establishes the implication.
Indirect proof (proof by contrapositive): A proof that proceeds by showing that if q is false then p must be false. Proving the contrapositive (¬q → ¬p) is logically equivalent to proving p → q.
Therefore, the descriptions pair as follows: vacuous proof with the description based on p being false; trivial proof with the description based on q being true; direct proof with the description that shows q from p; indirect proof with the description that shows p is false when q is false.