Problem:
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?
Answer Choices:
A. All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
B. No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
C. All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
D. All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
E. All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
Solution:
Consider a plot in which the number of T-shirts sold ( -axis) is graphed against the school size ( -axis) and each of the high schools is assigned a point on this plot. Let be the point assigned to Euclid HS. The statement given in the problem is that for every high school, if the -coordinate of its point on the plot is greater than , then the -coordinate of its point must be less than or equal to ; that is, "If , then ."
Statement (A) is equivalent to "If , then ."
Statement (B) is equivalent to "If , then ."
Statement (C) is equivalent to "If , then ."
Statement (D) is equivalent to "If , then ."
Statement (E) is equivalent to "If , then ."
Statement (B) is the contrapositive of the original statement and is therefore logically equivalent to it. All of the other choices have different meanings and can have different truth values. For example, assume and . If and , then the original statement is false, but statements (A), (D), and (E) are vacuously true (the hypothesis is false in each case). If and , then the original statement is true, but statement is false.
Let be the collection of schools bigger than Euclid HS. Let be the collection of schools that sold more T-shirts than Euclid HS. The original statement is that there is no that is also in . In other words, and are disjoint. Statement (B) says that there is no that is also in , which is also a way of saying that and are disjoint. Choice ( ) is incorrect because there might be a school bigger than Euclid HS that sold the same number of T-shirts as Euclid HS. Choices (A), (D), and (E) are incorrect because if all the other schools are bigger than Euclid HS and sold more T-shirts than Euclid HS, then the original statement is false but choices (A), (D), and (E) are (vacuously) true.
The problems on this page are the property of the MAA's American Mathematics Competitions