Deductive Logic, Syllogisms & Conditionals

Deductive logic requires identifying conclusions that necessarily follow from given premises.

• "All A are B": Set A is a subset of set B (A \subseteq B).
• "No A are B": Sets A and B are disjoint (A \cap B = \emptyset).
• "Some A are B": Set intersection contains at least one element (A \cap B \neq \emptyset).
• Conditional P \implies Q: Logically equivalent ONLY to its contrapositive \neg Q \implies \neg P.

1. Translate verbal premises into set diagrams or symbolic conditional logic (P -> Q).
2. Write down the contrapositive immediately (~Q -> ~P).
3. Select the option that MUST be true under all valid diagrams.