Final Exam Solutions
March 15, 2010
Problem 1
Part (a)
All we need to do here is translate the english denition into our language:
MetricSpace
=
cfw_S : type1 , d : S S R s.t. x : S.y : S. (d (x, y ) 0) (d (x, y ) = d (y, x)
(d (x, y ) = 0) (x = y ) z :

Midterm 2 Solutions
February 28, 2010
Problem 1
Using the notation dened in the problem statement:
f
= x : cfw_rst : int; second : real . cfw_rst x.second; second x.rst
g
= x : cfw_rst : real; second : int . cfw_rst x.second; second x.rst
Another acceptab

PS2, Solutions
February 7, 2010
Problem 1
The rst rule simply takes a production which gives a terminal symbol, and an edge which is labeled with the
corresponding symbol. It then labels the edge with the producing nonterminal:
X
nm
X
nm
The next rule bui

PS3 Solutions
February 25, 2010
Problem 1
By denition:
( g ) (x) (y )
1
(g (x + y ) g (x)
=
lim
=
lim
=
lim (yM x + xM y + yM y )
0
0
1
(x + y ) M (x + y ) xM x)
0
= yM x + xM y
g,
So in order to write an expression for
we must pull out the matrix
accompl

PS4 Solutions
March 14, 2010
Problem 1
Existence
d
For any v V , let
i=1 i (v ) xi be its expansion in the x1 , . . . , xd basis. We must show that these function
i : V R are linear (hence are elements of V ) and form a basis for V . First, note that i (x