Kenneth Rosen Edition 7 Exercise 6.2 Question 19 (Page No. 405 - 406)

Question

Suppose that every student in a discrete mathematics class of $25$ students is a freshman, a sophomore, or a junior.

• A. Show that there are at least nine freshmen, at least nine sophomores, or at least nine juniors in the class.
• B. Show that there are either at least three freshmen, at least $19$ sophomores, or at least five juniors in the class

Answer 1

FOR A

Suppose, by way of contradiction, that the statement is not true. Then there are not more than 8 freshmen, not more than 8 sophomores and not more than 8 juniors. This means that there are 24(8+8+8=24) students. However, we know that there are 25 students. This creates a contradiction. Thus, we know that the statement must be true.

FOR B

Suppose, by way of contradiction, that the statement is not true. Then there are not more than 2 freshmen, not more than 18 sophomores and not more than 4 juniors. This means that there are 24(2+18+4=24) students. However, we know that there are 25 students. This creates a contradiction. Thus, we know that the statement must be true.