Key Concepts and Formulas

• Contrapositive: The contrapositive of a statement "If p, then q" (denoted as $p \rightarrow q$) is "If not q, then not p" (denoted as $\sim q \rightarrow \sim p$).
• Negation of a conjunction: The negation of a conjunction "p and q" (denoted as $p \land q$) is "not p or not q" (denoted as $\sim p \lor \sim q$). This is De Morgan's Law.
• Negation of a subset: The negation of $A \subseteq B$ is $A \nsubseteq B$, which means A is not a subset of B.

Step-by-Step Solution

Step 1: Identify the given statement.

The given statement is: "If A $\subseteq$ B and B $\subseteq$ D, then A $\subseteq$ C". We can rewrite this using logical notation as:

$(A \subseteq B \land B \subseteq D) \rightarrow (A \subseteq C)$

Step 2: Find the negation of the conclusion.

The conclusion is $A \subseteq C$. The negation of this is $A \nsubseteq C$.

Step 3: Find the negation of the hypothesis.

The hypothesis is $A \subseteq B \land B \subseteq D$. The negation of this is $\sim (A \subseteq B \land B \subseteq D)$. Using De Morgan's Law, we have:

$\sim (A \subseteq B) \lor \sim (B \subseteq D)$, which is $(A \nsubseteq B) \lor (B \nsubseteq D)$.

Step 4: Formulate the contrapositive statement.

The contrapositive statement is "If not (A $\subseteq$ C), then not (A $\subseteq$ B and B $\subseteq$ D)". Using the negations we found in Steps 2 and 3, the contrapositive is:

If $A \nsubseteq C$, then $(A \nsubseteq B) \lor (B \nsubseteq D)$.

This statement can be written as: If A is not a subset of C, then A is not a subset of B or B is not a subset of D.

Step 5: Compare with the given options.

The derived contrapositive statement is: If $A \nsubseteq C$, then $A \nsubseteq B$ or $B \nsubseteq D$.

Now we check the options: (A) If A ⊈ C, then A ⊈ B or B ⊈ D (B) If A ⊈ C, then A ⊈ B and B $\subseteq$ D (C) If A $\subseteq$ C, then B $\subset$ A or D $\subset$ B (D) If A ⊈ C, then A $\subseteq$ B and B $\subseteq$ D

Option (A) matches our derived contrapositive statement. However, the stated "Correct Answer" is (C). Let's check if there's an error in the provided answer. Our derived contrapositive is logically equivalent to option (A).

The correct contrapositive is "If $A \nsubseteq C$, then $A \nsubseteq B$ or $B \nsubseteq D$."

The provided correct answer is option (C). Option (C) states: "If A $\subseteq$ C, then B $\subset$ A or D $\subset$ B"

The original statement is: (A $\subseteq$ B and B $\subseteq$ D) $\to$ (A $\subseteq$ C) The contrapositive is: $\sim$(A $\subseteq$ C) $\to$ $\sim$(A $\subseteq$ B and B $\subseteq$ D) Which is: If A ⊈ C, then A ⊈ B or B ⊈ D

The correct option is (A). However, we are told the correct answer is (C). This suggests there's an error in the question or the given correct answer. Since we must follow the instructions and work toward answer (C), there must be a misinterpretation. Let's re-examine the contrapositive definition.

The contrapositive of $p \to q$ is $\sim q \to \sim p$. $p = (A \subseteq B \land B \subseteq D)$ $q = (A \subseteq C)$ So, $\sim q = (A \nsubseteq C)$ And $\sim p = \sim (A \subseteq B \land B \subseteq D) = (A \nsubseteq B \lor B \nsubseteq D)$

So the contrapositive is: If $A \nsubseteq C$, then ($A \nsubseteq B$ or $B \nsubseteq D$). This corresponds to option (A).

There is an error in the question, the correct answer is option (A), not (C). We are instructed to arrive at the given answer (C), which we cannot.

Common Mistakes & Tips

• Carefully apply De Morgan's Laws when negating compound statements connected by "and" or "or".
• Remember that $A \nsubseteq B$ is the negation of $A \subseteq B$.
• Double-check the contrapositive statement to ensure the "if" and "then" parts are correctly negated and swapped.

Summary

We started with the given statement $(A \subseteq B \land B \subseteq D) \rightarrow (A \subseteq C)$. We then found the negation of the conclusion and the negation of the hypothesis using De Morgan's Law. Finally, we constructed the contrapositive statement: If $A \nsubseteq C$, then $A \nsubseteq B$ or $B \nsubseteq D$, which corresponds to option (A). However, the provided correct answer is option (C), indicating an error in the problem statement or the provided answer. We are supposed to arrive at answer (C), which is impossible given the correct contrapositive.

Final Answer

The correct contrapositive, based on standard logic, corresponds to option (A). However, the question states that the correct answer is (C). Therefore, there appears to be an error in the problem statement or the given correct answer. The final answer is \boxed{A}.