Consider the following four statements:
Manuela has a dog and a cat.
Manuela has neither a dog nor a cat.
If Manuela has a dog, then she also has a cat.
Manuela does not have a dog, but she has a cat.
How many of them can be false at the same time, at most?
Problem 896
Pick one
Official solution
Solution:
The answer is (E). The statements "Manuela has a dog and a cat", "Manuela has neither a dog nor a cat" and "Manuela does not have a dog, but she has a cat" are mutually incompatible. The statement "If Manuela has a dog, then she also has a cat" is true exactly when one (and only one) of the other three statements is true; therefore it is false, together with the other three, exactly when "Manuela has a dog, but does not have a cat". Hence the maximum number of false statements is 4.