Question

Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement “Some human beings are not cruel creatures” is FALSE, then which of the following set of statement(s) can be logically inferred with certainty?
(i)  All human beings are cruel creatures.
(ii)  Some human beings are cruel creatures.
(iii) Some creatures that are cruel are human beings.
(iv) No human beings are cruel creatures.
 

• A. only (i)
• B. only (iii) and (iv)
• C. only (i) and (ii)
• D. (i), (ii) and (iii)

Hint:

When a statement of the form “Some $A$ are not $B$” is {false}, its negation “\(\forall x\in A,\; x\in B\)” is {true}. Combine this with existence (does $A$ actually contain anything?) to decide “some $A$ are $B$” and related claims.

Answer 1

The Correct Option is D

Let $H$ = set of human beings, $C$ = set of cruel creatures. The given false statement is
\[ \text{“Some human beings are not cruel”} \;\equiv\; \exists x\,(x\in H \land x\notin C). \] Its being false means the negation is true:
\[ \neg\exists x\,(x\in H \land x\notin C) \;\equiv\; \forall x\,(x\in H ⇒ x\in C). \] Hence \(\boxed{H \subseteq C}\). Also, the stem explicitly implies humans exist (“Human beings are one among many creatures that inhabit the world”). Finally, we are told \(\exists\) cruel creatures (“some creatures are cruel”). Now test each claim: 
(i) All human beings are cruel creatures. This is precisely \(\forall x\,(x\in H⇒ x\in C)\). \(⇒\) True. 
(ii) Some human beings are cruel creatures. Since $H\neq\varnothing$ (humans inhabit the world) and $H\subseteq C$, there exists at least one $h\in H\cap C$. \(⇒\) True. 
(iii) Some creatures that are cruel are human beings. From (ii), the witness $h$ is both human and cruel, hence \(C\cap H\neq\varnothing\). \(⇒\) True. 
[4pt] (iv) No human beings are cruel creatures. This would be \(H\cap C=\varnothing\), contradicting (ii)/(iii). \(⇒\) False. 
Therefore, exactly (i), (ii), (iii) follow with certainty. 
\[ \boxed{\text{Answer: (D)}} \]