Which of the following statements is false?

The statement that all rodents are harmless is false because rodents are known carriers of various diseases that can pose significant health risks to humans. For instance, diseases such as hantavirus, leptospirosis, and salmonellosis can be transmitted through direct contact with rodents or their droppings, urine, or nesting materials. This highlights the importance of maintaining proper hygiene and sanitation practices in food handling and preparation areas to minimize the risk of contamination from rodents.

In comparison, the other statements are true: food exposed to rodents must indeed be thoroughly cleaned or discarded to prevent foodborne illnesses; rodents can carry diseases, which can impact public health significantly; and effective trapping methods exist for controlling rodent populations. Understanding these facts is crucial for food handlers to ensure food safety and protect the health of customers.