Problem:
Tom has a collection of snakes, of which are purple and of which are happy. He observes that:
all of his happy snakes can add,
none of his purple snakes can subtract, and
all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
Answer Choices:
A. Purple snakes can add
B. Purple snakes are happy
C. Snakes that can add are purple
D. Happy snakes are not purple
E. Happy snakes can't subtract
Solution:
To see that choices , and do not follow from the given informtion, consider the following two snakes that may be part of Tom's collection. One snake is happy but not purple and can both add and subtract. The second is purple but not happy and can neither add nor subtract. Then each of the three bulleted statements is true, but each of these choices is false.
To show that answer choice is correct, first observe that the third bulleted statement is equivalent to "Snakes that can add also can subtract." The second bulleted statement is equivalent to "Snakes that can subtract are not purple." The three bulleted statements combined then lead to the conclusion .
Let , and denote the statements that a snake is happy, is purple, can add, and can subtract, respectively.
Let denote "implies" and denote "not". Recall that an implication is logically equivalent to its contrapositive
Then the three bulleted statements can be exactly summarized as
Choice , which is
follows from the transitivity of the "implies" relation. However, choices , which is ; , which is ; , which is ; and , which is , do not follow from those three implications.
The problems on this page are the property of the MAA's American Mathematics Competitions