CSE 2315 
Discrete Structures
Homework 2: Predicate Calculus, Proof Techniques, and Recursion
CSE 2315 
Discrete Structures
Homework 2 Solutions  Fall 2010
Due Date: Oct. 7 2010, 3:30 pm
1.
a)
1
.
(
∀
x
)
R
(
x
)
hyp
2
.
(
∀
x
)(
∀
y
)(
R
(
y
)
→
P
(
x,y,z
))
hyp
3
.
(
∀
y
)(
R
(
y
)
→
P
(
z,y,z
))
2
ui
legal.
z
is not within the scope of a quantiﬁer for
z
.
4
. R
(
z
)
→
P
(
z,a,z
)
3
ui
illegal. all bound occurrences have to be replaced with
the same variable or constant.
5
. R
(
z
)
1
ui
legal.
z
is not within the scope of a quantiﬁer for
z
.
b)
1
.
(
∃
z
)
Q
(
x,z
)
hyp
2
.
(
∀
x
)(
∀
y
)(
Q
(
x,y
)
→
P
(
x
))
hyp
3
.
(
∀
y
)(
Q
(
x,y
)
→
P
(
x
))
2
ui
legal.
x
is not within the scope of a quantiﬁer for
x
.
4
. Q
(
x,z
)
1
ei
illegal.
z
occurs previously in the proof sequence. 
However, it causes no error here since its only use was
by the existential quantiﬁer that is removed.
5
. Q
(
x,z
)
→
P
(
x
)
3
ui
legal.
z
is not within the scope of a quantiﬁer for
z
.
c)
1
.
(
∀
x
)(
∃
y
)
P
(
x,y
)
hyp
2
.
(
∀
x
)(
P
(
x,a
)
→
S
(
a
))
hyp
3
.
(
∃
y
)
P
(
x,y
)
1
ui
legal.
x
is not within the scope of a quantiﬁer for
x
.
4
. P
(
x,a
)
3
ei
illegal.
a
occurs previously in the proof sequence.
5
. P
(
x,a
)
→
S
(
a
))
2
ui
legal.
x
is not within the scope of a quantiﬁer for
x
.
6
. S
(
a
)
4
,
5
mp
d)
1
.
(
∃
x
)
P
(
x
)
hyp
2
.
(
∀
y
)(
P
(
y
)
→
Q
(
y,z
))
hyp
3
. P
(
z
)
1
ei
illegal.
z
occurs previously in the proof sequence.
4
. P
(
z
)
→
Q
(
z,z
)
2
ui
legal.
z
is not within the scope of a quantiﬁer for
z
.
5
. Q
(
z,z
)
4
,
3
mp
6
.
(
∃
z
)
Q
(
z,z
)
5
eg
legal. always legal for same variable name.
e)
1
.
(
∀
x
)(
R
(
x
)
→
(
∃
y
)
Q
(
x,y
))
hyp
2
.
(
∀
z
)(
∀
y
)
Q
(
z,y
)
hyp
3
. R
(
z
)
→
(
∃
y
)
Q
(
z,y
)
1
ui
legal.
z
is not within the scope of a quantiﬁer for
z
.
4
.
(
∀
y
)
Q
(
z,y
)
2
ui
legal.
z
is not within the scope of a quantiﬁer for
z
.
5
. R
(
z
)
4
,
3
mt
6
.
(
∃
x
)
R
(
x
)
4
eg
legal.
x
does not occur in
5
.
2010 Manfred Huber
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Discrete Structures
Homework 2: Predicate Calculus, Proof Techniques, and Recursion
f)
1
.
(
∃
x
)
P
(
x
)
hyp
2
.
(
∀
x
)((
∃
y
)
P
(
y
)
→
Q
(
x,y
))
hyp
3
. P
(
y
)
1
ei
illegal.
y
occurs previously in the proof sequence.
4
.
(
∃
y
)
P
(
y
)
→
Q
(
x,y
)
2
ui
legal.
x
is not within the scope of a quantiﬁer for
x
.
5
.
(
∃
y
)
P
(
y
)
3
eg
legal. Always legal for same variable name.
6
. Q
(
x,y
)
5
,
4
mp
7
.
(
∀
x
)
Q
(
x,y
)
6
ug
legal.
x
in
6
was not derived using a wff that contained
x
as a free variable or derived it using ei.
8
.
(
∀
y
)(
∀
x
)
Q
(
x,y
)
7
ug
illegal.
y
in
7
was derived from a free variable (step
2
).
g)
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 Spring '08
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 Recursion, Natural number, Prime number, Manfred Huber, Discrete Structures Homework

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