Adapting a proof about irrational numbers For this problem, you will need to use the following fact which is proven elsewhere in this material. For every integer n, exactly one of the following three facts is true:
• n = 3k, for some integer k.
• n = 3k+1, for some integer k.
• n = 3k+2, for some integer k.
Prove that if n is an integer such that 3|n2, then 3|n.
Negating statements and proofs by contradiction.
(a) Give an English sentence that is equivalent to the negation of the following statement:
Among any set of 16 days chosen from the calendar, at least three of the chosen days fall on the same day of the week.
(b) Give a proof by contradiction of the following statement:
Among any set of 16 days chosen from the calendar, at least three of the chosen days fall on the same day of the week.
(c) Give an English sentence that is equivalent to the negation of the following statement:
Among any group of 1000 people, at least three of the people have the same birthday.
(d) Give a proof by contradiction of the following statement:
Among any group of 1000 people, at least three of the people have the same birthday.