Understanding Compound Propositions

A proposition is a declarative statement that is either true or false. A compound proposition is formed by connecting two or more simple propositions using logical connectives such as 'and' ($\land$), 'or' ($\lor$), 'not' ($\neg$), 'if...then' ($\to$), 'if and only if' ($\leftrightarrow$), or related phrases like 'unless', 'neither...nor'.

Analyzing Each Statement

• Statement A: "Unless it rains in time, harvest will not be good."
  • This can be rephrased using logical connectives: "If it does not rain in time, then the harvest will not be good."
  • This is a conditional statement ($\neg P \to \neg Q$), where P is "it rains in time" and Q is "harvest will be good".
  • Therefore, Statement A is a compound proposition.
• Statement B: "Only meritorious students will be considered for scholarship."
  • This statement defines a condition for scholarship consideration. It implies: "If a student is considered for scholarship, then they are meritorious."
  • While it has a conditional structure, it presents a single defining rule rather than combining two distinct, explicitly stated propositions with a connective. In many contexts, such statements are treated as simple propositions defining a property or class.
  • Therefore, Statement B is considered a simple proposition in this context.
• Statement C: "Be neither a borrower nor a lender."
  • This statement uses the 'neither...nor' construction. Logically, this is equivalent to "You are not a borrower AND you are not a lender."
  • This structure combines two propositions ("You are a borrower", "You are a lender") using negation and conjunction ($\neg P \land \neg Q$).
  • Therefore, Statement C is a compound proposition.
• Statement D: "Two triangles are formed if a square is divided diagonally."
  • This is a conditional statement: "If a square is divided diagonally, then two triangles are formed."
  • This uses the 'if...then' connective ($P \to Q$), where P is "a square is divided diagonally" and Q is "two triangles are formed".
  • Therefore, Statement D is a compound proposition.
• Statement E: "Not all good speakers are good writers."
  • This statement negates a universal claim ("All good speakers are good writers"). It asserts that there exists at least one good speaker who is not a good writer.
  • While it involves negation, it makes a single assertion about the relationship between the two qualities. It's not formed by combining two simpler propositions with connectives like 'and', 'or', or 'if...then'.
  • Therefore, Statement E is a simple proposition.

Conclusion

Based on the analysis, the compound propositions are A, C, and D.