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THE UNIVERSITY OF BRITISH COLUMBIA
Math 335 Section 201
Calculators are allowed
No cell phones or information sheets
Midterm begins at 11:00 am and ends at 11:50am
MIDTERM 2 March 6, 2013
NAME
STUDENT NUMBER
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1(a). The mystery set of A. Suppose S is the set defined by
S = {Monique, Robert, Lucy, Mark}. Consider the following pairing elements of S with elements of P(S):
Elements of S Elements of P(S)
Monique {Monique, Robert, Lucy}
Robert {Lucy, Mark}
Lucy {Lucy, Monique}
Mark {Lucy}
Suppose that each student in S is given the list of names shown above beside their name. The teacher says that those whose names are not on the list they received will be getting an A, otherwise they get a B. Who has received an A? Did anyone’s list contain exactly those students receiving an A? Did anyone’s list contain exactly those students receiving a B?
(b) Consideranyfunctionf from setA to its powerset P(A) (bydefinition, P(A) is the set of all the subsets ofA). LetM be the subset ofA consisting of all the elements x ofA such that x is not an element off(x).
Show that M cannot be of the form f(x) for any element x of A. (Hint: Assume that M = f(x) and prove that this leads to a contradiction).
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2. Pythagoras Word Problems
(a) Tony has got his kite stuck at the top of a very tall tree. He knows the string on his kite is 20 m long. When he pulls the string tight and holds the very end on the ground it touches 17 m from the bottom of the tree. If the ground is flat, how tall is the tree?
(b) Robert is using a 5m ladder to climb in his upstairs bedroom window. He finds that if he puts the base of the ladder 3.3m from the wall the top leans on the windowsill. How high from the ground is the windowsill?
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3. Consider the line segment below and assume it satisfies the equation
L/S = (L+S)/L
(a) Let L+S = 1 and L = x and then write the equation above as an equation in x.
(b) Rewrite the equation you have found in part (a) as a quadratic equation in x.
(c) Solve the quadratic equation you have found in part (b).
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4. (a)In Smallville there are 15 telephones. Can these be connected so that
there are 4 telephones each connected to 3 others, 8 telephones, each connected to 6 others, and 3 telephones, each connected to 5 others? Why or why not?
(b) In Verysmallville there are 7 phones. Can these be connected so that there are 3 phones connected to 4 others, 2 connected to 3 others and 2 connected to 2 others? If not explain why not. If so, draw a graph with these connections.
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5a.Which of these graphs has/have Euler circuit(s) ?
AGraph 3 only
BGraphs 1 and 3
CGraph 2 only
DGraph 1 only
ENone of the above
5b.Which of these graphs has an Euler path but does not have an Euler circuit ?
AGraph 3 only
BGraphs 1 and 3
CGraph 2 only
DGraph 1 only
ENone of the above