MATH 272 Final: MATH 272 Amherst S16M272Final
15 Feb 2019
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Final Exam, Math 220, Spring 2014
Thursday, May 15, 2014
Instructions: Do all twelve numbered problems. If you wish, you may also attempt the two
optional bonus questions. Show all work, including scratch work, on your exam paper. If
you need more space, use the back of any page. JUSTIFY YOUR ANSWERS. WRITE
LEGIBLY. NO CALCULATORS OR CELL PHONES.
1. (10 points). Let A,B,Cbe sets. Prove that
A∩B∩C⊆Cr(ArB)∪(BrA).
2. (15 points).
2a. (5 points). Write down the negation of the following statement:
∀a∈Z,∀b∈N,∃c∈Ns.t. ac > ab.
2b. (10 points). Prove the statement you gave as the answer to 2a above.
3. (20 points) Prove that [
n∈Nh1
n,3i= (0,3].
4. (20 points) For this problem, you may leave your numerical answers in terms
of factorials and basic arithmetic operations.
4a. (10 points) For a big party they are hosting, Alice and Bob decide to order 20 large
pizzas from a restaurant that serves 15 types of pizza. In how many ways could they place
this order?
4b. (10 points) At the end of the party, there are 10 slices of pizza left over: 2 cheese, 3
pepperoni, and 5 veggie. Alice and Bob tell the last eight guests to each take as many of the
slices home with them as they want. In how many ways could the guests do this, if together
they end up taking all the slices?
[Note: any two slices of the same type are identical; but of course, the eight guests are all
different people. Also, the taking of slices need not be fair; a single guest could take all ten
slices, for example.]
5. (15 points) For this problem, you may leave your numerical answers in terms
of factorials and basic arithmetic operations.
A bag contains several marbles: 6 blue marbles, 5 green marbles, and 3 red marbles.
5a. (8 points) If I pull two marbles out of the bag at random, what is the probability
that they are the same color?
5b. (7 points) If I pull three marbles out of the bag at random, what is the probability
that all three are different colors?
6. (20 points) Let g:R r {2} → Rbe the function given by the formula
g(x) = x+ 6
x−22
.
Prove that the image g(2,6)of the interval (2,6) is the interval (9,∞).
7. (10 points) Let f:R→Rbe a function. Define a function
h:R×R→R×Rby h(x, y)=y, f (x) + y2+ 5.
