SOLUTIONS TO HOMEWORK 5
MATH 150, FALL 09
Problem 1.
Section 2.2/ Exercise 8.
Suppose
Σ
is a set of formulas such that for all sentences
τ
, either
Σ

=
τ
or
Σ

=
¬
τ
. Assume that
A

= Σ
. Show that for any
τ
,
A

=
τ
iff
Σ

=
τ
.
For one of the directions, suppose that Σ

=
τ
.
Then by definition of
logical implication and since
A

= Σ, it follows that
A

=
τ
.
For the other direction, suppose that
A

=
τ
.
Then
A
6
=
¬
τ
.
By the
definiton of logical implication and since
A

= Σ, it follows that Σ
6
=
¬
τ
.
So, by the assumptions on Σ, we get that Σ

=
τ
.
Problem 2.
Section 2.2/ Exercise 9.
Assume that the language has equality and a twoplace predicate
P
. For
each of the following find a sentence
σ
such that the model
A
satisfies
σ
iff
the condition is met.
(1)
The universe of
A
has exactly two members.
(2)
P
A
is a function (from the universe of
A
to itself).
(3)
P
A
is a permutation of the universe of
A
(i.e.
it is a onetoone,
onto function).
(1)
σ
= “
∃
x
∃
y
(
¬
(
x
=
y
)
∧ ∀
z
(
z
=
x
∨
z
=
y
))”
(2)
σ
= “
∀
x
∃
y
(
P
(
x, y
)
∧ ∀
z
(
P
(
x, z
)
→
y
=
z
))”
(3) Let
σ
f
= “
∀
x
∃
y
(
P
(
x, y
)
∧∀
z
(
P
(
x, z
)
→
y
=
z
))” and
σ
11
= “
∀
x
∀
y
(
∃
z
(
P
(
x, z
)
∧
P
(
y, z
))
→
x
=
y
)” and
σ
o
= “
∀
y
∃
xP
(
x, y
)”. Then
σ
f
says that
P
A
is a function,
σ
11
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 Spring '09
 SMITH
 Math, Logic, Formulas, Deduction, Logical implication

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