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Chapter 2: Problem 23

Nach einem Mordfall gibt es drei Verd盲chtige, \(A\), \(B\) und \(C\), von denen zumindest einer der T盲ter sein muss. Nachdem sie und die Zeugen getrennt vernommen wurden, kennen die Ermittler folgende Fakten: 1\. Wenn \(A\) T盲ter ist, dann m眉ssen \(B\) oder \(C\) ebenfalls T盲ter sein. 2\. Wenn \(B\) T盲ter ist, dann ist \(A\) unschuldig. 3\. Wenn \(C\) T盲ter ist, dann ist auch \(B\) T盲ter. L盲sst sich damit herausfinden, wer von den dreien schuldig bzw. unschuldig ist?

Short Answer

Expert verified
Answer: Suspect A is innocent, while Suspects B and C are guilty.

Step by step solution

01

Fact 1:

If \(A\) is the culprit, then either \(B\) or \(C\) must also be the culprit (or even possibly both). This can be represented as: If A, then (B or C).
02

Fact 2:

If \(B\) is the culprit, then \(A\) is innocent. This can be represented as: If B, then not A.
03

Fact 3:

If \(C\) is the culprit, then \(B\) is also the culprit. This can be represented as: If C, then B. Now, let's use these facts to determine who is guilty and who is innocent.
04

Analyzing Fact 2:

From Fact 2, we know that if B is guilty, then A is innocent. Since at least one of the three suspects must be guilty, if we know that \(A\) is innocent, then either \(B\) or \(C\) must be guilty. So, if Fact 2 is true, we only need to focus on the possible cases with \(B\) and \(C\) as the culprits.
05

Analyzing Fact 3:

Fact 3 tells us that if \(C\) is a culprit, then \(B\) is also a culprit. However, if \(B\) is a culprit, we know from Fact 2 that \(A\) must be innocent. Since at least one of them must be guilty, and we've already established that if \(B\) is guilty, then A is innocent, this means that if Fact 3 is true, both \(B\) and \(C\) must be guilty.
06

Analyzing Fact 1:

Recall that Fact 1 tells us if A is guilty, then either B or C must also be guilty. Now, we already know from the analysis of Fact 2 and Fact 3 that if B and C are guilty, A must be innocent. Therefore, A cannot be guilty, since it would contradict the conclusions from Fact 2 and Fact 3.
07

Conclusion:

Based on the analysis of the given facts, we can conclude that Suspect A is innocent, while Suspects B and C are guilty.

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Most popular questions from this chapter

Beweisen Sie die 脛quivalenzen: $$ \begin{aligned} (A \vee B) & \Leftrightarrow \neg(\neg A \wedge \neg B) \\ (A \wedge B) & \Leftrightarrow \neg(\neg A \vee \neg B) \\ (A \Rightarrow B) & \Leftrightarrow((\neg A) \vee B) \end{aligned} $$

Die Menge \(A_{4}\) hat vier Elemente, die Mengen \(B_{3}\), \(B_{4}\) und \(B_{5}\) haben entsprechend drei, vier und f眉nf Elemente. 脺berlegen Sie jeweils, ob es Abbildungen $$ \begin{aligned} &f_{43}: A_{4} \rightarrow B_{3} \\ &f_{44}: A_{4} \rightarrow B_{4} \\ &f_{45}: A_{4} \rightarrow B_{5} \end{aligned} $$ geben kann, die (a) injektiv, aber nicht surjektiv, (b) surjektiv, aber nicht injektiv, (c) bijektiv sind.

Welche der folgenden Schl眉sse sind auf formaler Ebene (d. h. noch ohne tats盲chliche Betrachtung der Wahrheitswerte der Aussagen) richtig? Welche sind als Implikationen wahre Aussagen, wenn man auch die Wahrheitswerte der jeweils verknuipften Aussagen betrachtet? 1\. Alle V枚gel k枚nnen fliegen. M枚wen sind V枚gel. \(\Rightarrow\) M枚wen k枚nnen fliegen. 2\. Alle V枚gel k枚nnen fliegen. Pinguine sind V枚gel. \(\Rightarrow\) Pinguine k枚nnen fliegen. 3\. Alle V枚gel k枚nnen fliegen. M枚wen k枚nnen fliegen. \(\Rightarrow\) M枚wen sind V枚gel. 4\. Alle V枚gel k枚nnen fliegen. Libellen k枚nnen fliegen. \(\Rightarrow\) Libellen sind V枚gel.

Beweisen Sie die Abtrennregel (modus ponens): $$ (A \wedge(A \Rightarrow B)) \Rightarrow B $$

Ist der folgende Schluss richtig? (,,Wer von der Quantenmechanik nicht schockiert ist, der hal sie nicht verstanden" (Niels Bohr) \(\wedge\),Niemand versteht die Quantenmechanik" (Richard Feynman)) \(\Rightarrow\),Niemand ist von der Quantenmechanik schockiert"
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