quiz3 - 6 chessboard with three corners removed using 1 3...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Name: Math 55 Quiz 3 July 1, 2009 GSI: Rob Bayer You have until 4:00 to complete this quiz. You must show your work. 1. (3 pts) Prove that the function f : R R given by f ( x ) = 3 x - 4 is a bijection. (injective) If f ( x ) = f ( y ) , then 3 x - 4 = 3 y - 4 so adding 4 gives 3 x = 3 y and thus x = y . Therefore, f ( x ) = f ( x ) x = y and f is injective. (surjective) Let y R . Then y +4 3 R and f ( y +4 3 ) = 3 y +4 3 - 4 = y , so y xf ( x ) = y and f is surjective. f is a bijection 2. (3 pts) Let f : D E be a function and let A,B D . Prove that f ( A B ) f ( A ) f ( B ) (Proof 1) Let y f ( A B ). So x A B such that f ( x ) = y . Since x A B , x A , so y = f ( x ) f ( A ). Similarly, x B so y f ( B ). Thus, y f ( A ) f ( B ). (Proof 2) A B A so f ( A B ) f ( A ). Similarly, f ( A B ) f ( B ). Thus, f ( A B ) f ( A ) f ( B ) 3. (4 pts) Show that it is not possible to tile a 6
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6 chessboard with three corners removed using 1 3 rectangles Well color the board as follows: X R G B R X G B R G B R R G B R G B B R G B R G G B R G B R R G B R G X Note that anywhere you put a 1 3 triominoe on this board, it will cover exactly one red, one green, and one blue square. Since there are 33 squares to cover, any covering must use 11 triominoes. However, a quick count shows that there are 12 red squares, and thus they cannot all be covered by just 11 triominoes. Thus, there is no possible tiling of this board....
View Full Document

Ask a homework question - tutors are online