# Logic problem - what kind of logic is it?

I would be most gratefull, if someone could verify my solution to this problem.

 1) Of Aaron, Brian and Colin, only one man is smart.

Aaron says truthfully:
1. If I am not smart, I will not pass Physics.
2. If I am smart, I will pass Chemistry.

Brian says truthfully:
3. If I am not smart, I will not pass Chemistry.
4. If I am smart, I will pass Physics.

Colin says truthfully:
5. If I am not smart, I will not pass Physics.
6. If I am smart, I will pass Physics.

While
I. The smart man is the only man to pass one particular subject.
II. The smart man is also the only man to fail the other particular subject.

Which one of the three men is smart? Why?


I would say that it could have been any one of them, as the implications in every statement are not strong enough to disprove the statements I,II. But I'm not sure if my solution is enough, as I'm not sure what kind of logic it is.

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Well, if A is smart, then B failed Chemistry, C failed physics, so A can't fail neither Chemistry or Physics (since Mr Smart fails one subject as the only one). If C is smart, A fails Physics, B fails Chemistry, reasoning same as above. Hence, only possible one is that B is smart, he passes Physics, but fails Chemistry. –  H. M. Šiljak May 17 '11 at 18:13

Brian must be the smart man. For if Aaron is smart he passes Chemistry and so must be the only one to fail Physics, but Colin fails Physics. Likewise, if Colin is smart he passes Physics and must be the only one to fail Chemistry, but Brian fails Chemistry. Only Brian does not lead to a similar contradiction.

This is not much of a logic-based solution since simple elimination is enough here; I believe you can formalize it with standard propositional calculus.

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Let use:
$X_S$ to mean $X$ is smart. $X_{\bar S}$ to mean $X$ is not smart.
$X_P$ to mean $X$ passes physics. $X_{\bar P}$ to mean $X$ does not pass physics.
$X_C$ to mean $X$ passes Cemistry. $X_{\bar C}$ to mean $X$ does not pass Chemistry.

1) Of Aaron, Brian and Colin, only one man is smart. translates to : $A_S\lor B_S \lor C_S \equiv T$
$(A_S \land B_S) \lor (A_S \land C_S) \lor (B_S \land C_S) \equiv F$

What Aaron,Brian and Colins said truthfully can be stated as:

$\left( A_S \land A_C \land A_{\bar P}\right ) \lor \left( A_{\bar S} \land A_{\bar P} \right ) \tag{I}$
$\left( B_S \land B_{\bar C} \land B_P \right ) \lor \left( B_{\bar S} \land B_{\bar C} \right ) \tag{II}$
$\left( C_S \land C_{\bar C} \land C_P\right ) \lor \left( C_{\bar S} \land C_{\bar P} \right ) \tag{III}$

While 1. The smart man is the only man to pass one particular subject. can be stated as $X_S \land X_{P} \land Y_{P} \equiv X_S \land X_{C} \land Y_{C} \equiv F$
2. The smart man is also the only man to fail the other particular subject. $X_S \land X_{\bar P} \land Y_{\bar P} \equiv X_S \land X_{\bar C} \land Y_{\bar C} \equiv F$

$I \land II \land III \equiv B_S \land B_P \land B_{\bar C} \land A_{\bar S} \land A_{\bar P} \land C_{\bar S} \land C_{\bar P} \tag{IV}$

Which one of the three men is smart? Why? Brian is the smart one because of $IV$.

I think this is simple case of combinatorial logic.

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Here is an example:

Suppose Aaron is the smart one. This means Brian and Colin are not smart. Thus Aaron passes Chemistry (by 2) and fails Physics (by II), but Colin also failed Physics (by 5), contradicting II. Thus Aaron is not the smart one.

So, see if you can finish the problem using this kind of reasoning.

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You could just create a simple table and read the solution:

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