Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

476 Chapter 9. Finite and Infinite Sets
where p 1 , p 2 ,...,p r are distinct prime numbers, q 1 , q 2 ,...,q s are distinct
prime numbers, and ˛1, ˛2,..., ˛r and ˇ1, ˇ2,..., ˇs are natural numbers.
We also write 1 D 2 0 when m D 1. We then define
If x D m 1
f.x/ D p 2˛1
1
p 2˛2
2
p 2˛r r q 2ˇ1 1
1
q 2ˇ2 1
2
q 2ˇs 1
s :
, then we define f.x/ D p2˛1
1
p 2˛2
2
p 2˛r r D m 2 .
2 5
(a) Determine f , f , f.6/, f.
3
6
12 375 2 3
25 /, f 11 3
,andf
392 3 5 4 .
(b) If possible, find x 2 Q C such that f.x/ D 100.
(c) If possible, find x 2 Q C such that f.x/ D 12.
(d) If possible, find x 2 Q C such that f.x/ D 2 8 3 5 13 17 2 .
(e) Prove that the function f is an injection.
(f) Prove that the function f is a surjection.
(g) What has been proved?
9.3 Uncountable Sets
Preview Activity 1 (The Game of Dodge Ball)
(From The Heart of Mathematics: An Invitation to Effective Thinking by Edward
B. Burger and Michael Starbird, Key Publishing Company, c2000 by Edward B.
Burger and Michael Starbird.)
Dodge Ball is a game for two players. It is played on a game board such as
the one shown in Figure 9.4. Player One has a 6 by 6 array to complete and Player
Two has a 1 by 6 row to complete. Each player has six turns as described next.
Player One begins by filling in the first horizontal row of his or her table with
a sequence of six X’s and O’s, one in each square in the first row.
Then Player Two places either an X or an O in the first box of his or her row.
At this point, Player One has completed the first row and Player Two has
filled in the first box of his or her row with one letter.