Math 461 (Spring 2018) – Practice Midterm 2
The following is meant as a practice version of Midterm 2. The real Midterm 2, on
Monday,
April 16, in class
, will be similar, but shorter.
You only need to use the “official” syntax for
terms and formulas in the first two problems.
1.
Let
L
be a language consisting of the constant symbol
0
, a binary function symbol +, and a
binary relation symbol
<
. In following formula
ϕ
:
(
∀
v
0
∃
v
1
<
0
+
v
0
v
1
→ ∃
v
2
(
<
0
+
v
2
v
2
∧ ¬
< v
1
v
2
))
write
ϕ
out “informally” (without using prefix/Polish notation, adding in parentheses, etc), and
identify all of the (i) terms, (ii) atomic formulas, (iii) free variables, and (iv) bound variables.
Solution:
Informally, we can write this formula as
(
∀
x
∃
y
(
0
< x
+
y
))
→
(
∃
z
(
0
< z
+
z
∧
y
6
< z
))
The (i) terms in (the formal version of)
ϕ
are:
v
0
,
v
1
,
v
2
,
0
,
+
v
0
v
1
,
+
v
2
v
2
.
The (ii) atomic formulas in
ϕ
are:
<
0
+
v
0
v
1
,
<
0
+
v
2
v
2
,
< v
1
v
2
.
The (iii) free variables in
ϕ
are:
v
1
.
The (iv) bound variables in
ϕ
are:
v
0
,
v
1
,
v
2
.
2.
Let
L
be a language
L
.
(a) Give the generating family for the set of all
L
-terms, and state the corresponding induction
principle (i.e., “induction on
L
-terms”).
(b) Given an
L
-structure
M
, and a variable assignment
s
in
M
(with domain
M
), prove by induc-
tion on
L
-terms that
s
(
t
)
∈
M
for all
L
-terms
t
. [Typo in old version: It should be
s
(
t
), not
s
(
t
).]
Solution:
(a) The set of
L
-terms, Term
L
, is generated by
(Exp
L
,
Var
∪ C
,
{F
f
}
f
∈F
)
where Exp
L
is the set of all
L
-expressions (i.e., finite sequences of symbols from the language),
Var =
{
v
n
:
n
∈
N
}
is the set of all variables,
C
is the set of constant symbols in the language, and
for each function symbol
f
∈ F
in the language, say with arity
k
,
F
f
: Exp
k
L
→
Exp
L
is given by
F
f
(
t
1
, . . . , t
k
) =
ft
1
. . . t
k
.
[the concatenation of these strings of symbols]