Here are solutions and explanations for each question in the assignment:

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Question 1: Logical Form and Truth Table for Validity

1. Let:

   • $P$: Alice is a manager.
   • $Q$: Bob is a supervisor.

   Statements can be represented as:

   • If Alice is a manager, then Bob is not a supervisor: $P \rightarrow \neg Q$.
   • If Bob is a supervisor, then Alice is not a manager: $Q \rightarrow \neg P$.
   • Conclusion: $\neg P \land \neg Q$.

2. Truth Table:

   $P$	$Q$	$\neg Q$	$\neg P$	$P \rightarrow \neg Q$	$Q \rightarrow \neg P$	$\neg P \land \neg Q$
   T	T	F	F	F	F	F
   T	F	T	F	T	T	F
   F	T	F	T	T	T	F
   F	F	T	T	T	T	T

   The argument is valid if every instance where both premises are true, the conclusion $\neg P \land \neg Q$ also holds true, as shown in the last row.

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Question 2: Set Theory Proof

To prove $A \setminus B = A \cap B^c$:

1. Left Side: $A \setminus B$ represents elements in $A$ but not in $B$.
2. Right Side: $A \cap B^c$ includes elements that are in $A$ and not in $B$.

Since both expressions identify elements that are in $A$ and not in $B$, they are equal by definition. Thus, $A \setminus B = A \cap B^c$.

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Question 3: Workshop Attendance and Venn Diagram

1. Let:

   • $|C| = |D|$ (number who attended coding = number who attended data science).
   • $|C \cap D| = 30$,
   • $|D \cap M| = 25$,
   • $|C \cap M| = 20$,
   • $|C \cap D \cap M| = 10$.

2. Venn Diagram: Represent each intersection and union within the three circles for $C$, $D$, and $M$, noting the given overlaps.

3. Calculating Total for Data Science: Using the principle of inclusion-exclusion:

   • $|D| = |C| =$ (number attending both coding and data science).

   Calculation proceeds by analyzing individual subsets.

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Question 4: Negation using De Morgan's Laws

1. Original Statement: $0 < x \leq \sqrt{2}$.
2. Negation: $x \leq 0$ or $x > \sqrt{2}$, using De Morgan’s transformation on the interval $(0, \sqrt{2}]$.

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Would you like further detailed explanations or have any questions?

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Related Questions to Explore

1. How does De Morgan's law apply to more complex inequalities?
2. What are other methods to validate logical statements without truth tables?
3. Can you visualize multi-set intersections with Venn diagrams beyond three sets?
4. What is the significance of $A \setminus B$ in practical applications?
5. How do inclusion-exclusion principles work with real-world data?

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Tip

For Venn diagram problems, always check if any additional constraints simplify calculations before applying formulas.