Question:hard

Assume that a Creative (\(C\)) person will Succeed (\(S\)) if the person is also Disciplined (\(D\)), but will not succeed otherwise. Now, consider the following statements:
(i) \(C \wedge S \Leftrightarrow D\)
(ii) \(C \Rightarrow (S \Leftrightarrow D)\)
(iii) \(C \Leftrightarrow ((D \Rightarrow S) \vee \neg S)\)
Which of the following options is correct?

Show Hint

Rewrite the given rule as C => (S <=> D), then test each statement against the valid (C,S,D) combinations.
Updated On: Jul 22, 2026
  • Both (i) and (ii) are TRUE
  • Only (ii) is TRUE
  • Both (ii) and (iii) are TRUE
  • Only (iii) is TRUE
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: Write the given rule as a formula.
"A Creative person succeeds if and only if disciplined" (with succeeding being impossible for creative-but-undisciplined people) translates to the axiom $A: C \Rightarrow (S \Leftrightarrow D)$. Any world we consider must make $A$ true, since it is given as a fact about how these three properties relate.

Step 2: Go through all 8 combinations of $C, S, D$ and mark which satisfy $A$.
$C{=}0$: $A$ is automatically true no matter what $S, D$ are, since an implication with a false antecedent is always true. That covers all 4 rows with $C{=}0$.
$C{=}1$: $A$ needs $S \Leftrightarrow D$, so only $(S{=}0,D{=}0)$ and $(S{=}1,D{=}1)$ satisfy it; the rows $(S{=}0,D{=}1)$ and $(S{=}1,D{=}0)$ with $C{=}1$ violate $A$ and are excluded.
So the 6 surviving rows are: $(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,1,1)$, written as $(C,S,D)$.

Step 3: Check statement (ii) is just $A$ itself.
Statement (ii) is $C \Rightarrow (S \Leftrightarrow D)$, which is identical to $A$. Since we only ever consider worlds where $A$ holds, statement (ii) is true in every one of the 6 surviving rows, so it is always TRUE.

Step 4: Check statement (i) on the row $(0,0,1)$.
Here $C \wedge S = 0 \wedge 0 = 0$, while $D = 1$. So $C \wedge S \Leftrightarrow D$ becomes $0 \Leftrightarrow 1$, which is false. Statement (i) fails on this valid row, so it is not always true.

Step 5: Check statement (iii) on the row $(0,0,0)$.
Here $D \Rightarrow S = 0 \Rightarrow 0 = 1$ and $\neg S = 1$, so $(D \Rightarrow S) \vee \neg S = 1$. But $C = 0$, so $C \Leftrightarrow 1$ is false. Statement (iii) fails on this valid row too, so it is not always true.

Final Answer:
Only statement (ii) is guaranteed true across every world consistent with the rule, so the answer is option (B).
Was this answer helpful?
0