(b) Give a recursive definition of the sequence {a
n
}, where
n = 1,2,3,.. if a
n
= 2n + 1. [Be very careful of the
quantification used in the induction step.]
Basis Step:
Induction Step:
(c) Give a recursive definition of the set S of positive
integers that are multiples of 3.
Basis Step:
Induction Step:
4. (10 pts.) Let {a
n
} be defined by the formula a
n
= 5n + 2 for
n = 1,2,3,
....
Define the sequence {b
n
} recursively by b
1
= 7 and
b
n+1
= b
n
+ 5 for n = 1,2,3,
....
Give a proof by induction that
a
n
= b
n
for n = 1,2,3,
....
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TEST2A/MAD2104
Page 3 of 4
5. (15 pts.)
Label each of the following assertions with "true"
or "false".
Be sure to write out the entire word.
(a) The set of all subsets of the natural numbers is countable.
(b) The number of onetoone functions from a set with 8
elements to a set with 15 elements is P(15,8).
(c) The coeffient of x
3
y
5
in the expansion of (x + y)
8
is the
number of 3 element subsets of an 8 element set.
(d) There is a onetoone correspondence between the natural
numbers and the real numbers in the interval (0,1).
(e) The total number of ways to assign truth values to five
truefalse problems by using the letters
T
and
F
is given by the
sum below.
5
C(5,k)
k=0
6. (10 pts.)
What is the minimum number of students required in
your Discrete Mathematics class to be sure that at least 4 have
birthdays occurring on the same day of the week this year?
Explain.
TEST2A/MAD2104
Page 4 of 4
7. (10 pts.)
(10 pts.) The following proposition represents an
invalid argument form:
[¬p
∧
(p
→
q)]
→
¬q.
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 Spring '08
 STAFF
 Logic, pts, Natural number, induction step

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