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Exam 2 review topics
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MATH 1313
2.7 Decimal expansion of real numbers (rationals and irrationals)
 Be able to use the decimal expansion of a number to locate it on the number line
 Know the distribution of the rational and irrational numbers: between any two
real numbers, you can always find a rational number and between any two real
numbers you can always find an irrational number.
 How do you tell a rational from an irrational? An irrational number has a decimal
expansion that never terminates and is not periodic (doesn’t eventually repeat.) A
rational number’s decimal expansion will either terminate or become eventually
periodic.
 Given a terminating or periodic decimal expansion, be able to write it as a
fraction. (i.e. write 12.34509090909… as a fraction)
 Be able to find an irrational number between any two other given numbers. (i.e.
find an irrational number between 0.134246 and 0.134247)
3.1, 3.2, 3.3 Are all infinities the same?
 Know what a one-to-one correspondence is.
 Know the definition of the cardinality of a set (the number of objects in the
collection – this can be finite or infinite)
 Know what it means for two sets to have the same cardinality (i.e. the same
number of objects in them), that is there is a one-to-one correspondence between
the members of the two sets. Every object in one set gets paired with an object in
the other set and vice versa.
 Be able to find a one-to-one correspondence between the natural numbers and
o The even numbers
o The odd numbers
o All natural numbers except the numbers 1 through 10
o All rational numbers
o The perfect squares
o Etc.
 The real numbers and the natural numbers do not have the same cardinality, that
is, you cannot list all the real numbers without leaving any out of the list.
 Be able to explain Cantor’s diagonalization proof that the set of real numbers has
strictly larger cardinality than the set of natural numbers. If I give you a list of
real numbers, you should be able to build a number that is not in the list.
8.4 Voting Theory
 Know that for an election with only two candidates, Majority Rule is a perfectly
fair voting method.
 Using a preference schedule from an election, be able to find the winner using
Plurality, Borda Count, Plurality with Elimination, or Sequential Pairwise with
Agenda method.
 Be able to determine if an election has a Condorcet winner


Be able to explain the four fairness criteria for a voting method that we discussed
– Go along with consensus, Condorcet Criterion, Independence of Irrelevant
Alternatives (ignore the irrelevant), and Monotonicity (better is better). And be
able to recognize a example of when a voting system violates one of the fairness
criterion.
Be able to explain the gist of Arrow’s Impossibility Theorem – i.e. when there are
three or more alternatives, it is impossible to devise any voting method in a way
that satisfies some basic fairness principles, so devising a flawless voting scheme
when there are three or more candidate is impossible.
Sample problems
1. Write the following numbers as fractions if it is possible:
a. 7.12344444444….
b. 0.000101001000100001000001000000100000001…
c. 0.234234234234…
d. 12.234567
e. √2
2. Find a rational number between
a. 1.414213 and √2
b. 2.34111111….. and 2.341111116
c. -5.231 and -5.2301
3. Find an irrational number between
a. 1.414213 and √2
b. 2.34111111….. and 2.341111116
c. -5.231 and -5.2301
4. List the positive rational numbers in one list so that the pattern of the order is
clear and so that all the positive rational numbers would eventually appear on the
list.
5. Let S be the set of square roots of all positive integers. (So S={√0, √1, √2, √3,
√4,√5, √6, √7, …}) Prove that S has the same cardinality as N, the set of natural
numbers. (To receive full credit, you must include all details.)
6. Let S be the set of all real numbers between 0 and 1 with the property that their
decimal expansions only have 0’s and 7’s. For example, the following numbers
are elements of S:
0.7777007707070777707000…
0.000000000700007777700000007…
a. Show that there is a rational number in S.
b. Show that there is an irrational number in S.
c. Show that the cardinality of S is not equal to the cardinality of the set
of natural numbers.
d. Bonus: Does the cardinality of S equal the cardinality of the set of real
numbers? Make a guess; no justification required.
7. Here are the vote tallies from an election. Determine the winner using plurality,
plurality with elimination, and Borda count voting methods. Explain your
reasoning in each case.
Candidate A
Candidate B
Candidate C
First-choice votes
5
6
4
Second-choice votes 4
2
9
Third-choice votes
6
7
2
8. A 17-member committee must elect one of four candidates: R,S,T, or W. Using
the below preference schedule, determine the plurality winner. Could those
members who most prefer W vote strategically in some way to change the
outcome in a way that will benefit them?
6 voters
4 voters
3 voters
4 voters
1st place
R
S
T
W
nd
2 place
S
R
S
T
rd
3 place
T
T
R
S
4th place
W
W
W
R