n
=
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TEST2/MAD2104
Page 3 of 4
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4. (15 pts.)
(a) Suppose f is a function defined recursively
by f(0) = 1 and f(n+1) = 2f(n) for n = 0,1,2,3,
....
Then
f(1) =
f(2) =
f(3) =
f(4) =
(b) Give a recursive definition for the sequence {a
n
},
where n = 1,2,3,..., if a
n
= 6n. [Be very careful with the
quantification used in the induction step.]
Basis Step:
Induction Step:
(c) Give a recursive definition of the set S of positive
integers that have a remainder of 2 upon division by 3. Be very
careful with the quantification in the induction step.
Basis Step:
Induction Step:
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6. (10 pts.)
Use mathematical induction to prove that a set
with n elements has n(n1)/2 subsets containing exactly two
elements whenever n is a positive integer greater than or equal
to 2.
TEST2/MAD2104
Page 4 of 4
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7. (10 pts.)
Using a complete sentences, correctly name the
fallacy exemplified by each of the following invalid arguments.
(a) "If x is any real number with x > 3, then x
2
> 9.
Suppose that x
≤
3. Then x
2
≤
9."
(b) "If x is any real number with x > 3, then x
2
> 9.
Suppose that x
2
> 9. Then x > 3."
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8. (15 pts.)
This is a valid argument: "All dogs are
carnivorous. Some animals are dogs. Therefore some animals are
carnivorous." The validity of the argument can be seen easily by
symbolizing the argument using propositional functions and
quantifiers as follows:
Define propositional functions as follows:
D(x) :
"x is a dog."
C(x) :
"x is carnivorous."
A(x) :
"x is an animal."
Then the argument translates into this:
(
∀
x)(D(x)
→
C(x))
(
∃
x)(A(x)
∧
D(x))
∴
(
∃
x)(A(x)
∧
C(x))
In the proof of validity which follows
provide the missing justification(s) for each step. You should
explicitly cite rules of inference and quantification,
hypotheses, and preceding steps by number. [Note: The lower case
latin letters ’a’,’b’,’c’ denote individuals in the universe of
discourse.]
1.
(
∀
x)(D(x)
→
C(x))
Justification:
Hypothesis
2.
(
∃
x)(A(x)
∧
D(x))
Justification:
Hypothesis
3.
A(c)
∧
D(c)
Justification:
4.
D(c)
→
C(c)
Justification:
5.
D(c)
∧
A(c)
Justification:
3,
∧
is commutative.
6.
D(c)
Justification:
7.
C(c)
Justification:
8.
A(c)
Justification:
9.
A(c)
∧
C(c)
Justification:
10. (
∃
x)(A(x)
∧
C(x))
Justification:
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 Spring '08
 STAFF
 Logic, pts, Natural number, induction step

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