If some Cops are Robbers, and no Robbers are Thieves, which of the following statements must be false?

To analyze the statements given the premises "Some cops are robbers" and "No robbers are thieves," we need to understand the implications of these premises logically.

1. ### Break Down the Problem

• Premise 1: Some cops are robbers.
• Premise 2: No robbers are thieves.
• We need to examine which statements cannot be true based on the above premises.

2. ### Relevant Concepts

• In logical terms, if "Some cops are robbers," it indicates that there is at least one individual who is both a cop and a robber.
• The statement "No robbers are thieves" indicates a mutual exclusivity between robbers and thieves.

3. ### Analysis and Detail

• Let's denote:
  • $C$: Set of cops
  • $R$: Set of robbers
  • $T$: Set of thieves
• According to the premises, we can represent them in the following way:
  • $C \cap R \neq \emptyset$ (Some cops are robbers)
  • $R \cap T = \emptyset$ (No robbers are thieves)

4. ### Verify and Summarize

• Any statement asserting that a cop is also a thief would contradict the premises. Therefore, any statement affirming the existence of an individual who is both a cop and a thief must be false.

Final Answer

Based on the premises provided, any statement claiming that "Some cops are thieves" or that "There exists at least one cop who is a thief" must be false.