1.3. Read the proof below of the theorem "if a class has 13 students, then two or more students
have their birthdays in the same month." and answer the questions that follow.
Proof by contradiction: Suppose that the class has 13 students and no 2 or more students have their
birthdays in the same month. That is PA Q is true. We are going to work on this assumption until
we obtain a contradiction. Since each student is born in some month, and since we are assuming that
no 2 or more students share the same birth month, there must be 13 or more months represented as
the birth months of the students in class. This is a contradiction. It contradicts a well-known fact that
there are only 12 months. Therefore we conclude that the assumption above is false. That is PA 10
is false which implies that PQ is true by the fact above. That is in a class of 13 students, 2 or
more students must share the same birth month.
1.3.1. Give the hypothesis and conclusion of this theorem.
1.3.2. Pick a statement that is a contradiction in the above proof and write what has been
contradicted.
1.3.3. Start the proof of the theorem directly.
(2)
(1)
1.3.4. If one has to prove the above theorem by a method of contrapositive which statement would
one prove?
1.3.5. Write the converse and the opposite of the above theorem.
(2)
(4)
