Math 557 Midterm Exam #2
November 4, 2005
SOLUTIONS
1. Let M = (UM , fM , gM , IM ) where UM = cfw_0, 1, 2, 3, 4, IM is the identity
relation on UM , and fM , gM are the binary operations of addition and
multiplication modulo 5. Thus M is essentially just
Math 557 Midterm Exam #1
October 8, 2003
1. Find a sentence in prenex normal form which is logically equivalent to
(x y Rxy ) x P x.
2. Using the predicates Bx (x is a barber in Podunk) and Sxy (x
shaves y ), translate the following argument into a senten
Math 557 Midterm Exam #2
November 10, 2003
1. Construct an LH -derivation of x (P x y P y ).
2. Let A be the logically valid sentence x (P x y P y ). Find a companion sequence C1 , . . . , Cn for A such that (C1 & & Cn ) A is a
quasitautology.
3. Let M =
Math 557 Midterm Exam #1
September 30, 2005
1. Find a sentence in prenex normal form which is logically equivalent to
(x y Rxy ) x P x.
2. Using the predicates Bx (x is a barber in Podunk), Cx (x is a
citizen of Podunk), and Sxy (x shaves y ), translate t
Math 557 Midterm Exam #2
November 4, 2005
4 problems
1. Let M = (UM , fM , gM , IM ) where UM = cfw_0, 1, 2, 3, 4, IM is the identity
relation on UM , and fM , gM are the binary operations of addition and
multiplication modulo 5. Thus M is essentially jus
Math 557 Homework #5
September 22, 2003
SOLUTIONS
1. Use signed tableaux to show that the following are logically valid.
(a) (x (A B ) (x A) (x B )
F (x (A B ) (x A) (x B )
T x(A B )
F (x A) (x B )
T x A
F x B
F B [x/a]
T A[x/a]
T (A B )[x/a]
/
\
F A[x/a]
Math 557 Midterm Exam #1
October 8, 2003
SOLUTIONS
1. Find a sentence in prenex normal form which is logically equivalent to
(x y Rxy ) x P x.
Solution. x y z ( Rxy ) P z ), or x y ( Rxy ) P y ).
2. Using the predicates Bx (x is a barber in Podunk) and Sx
Math 557 Homework #7
October 22, 2003
SOLUTIONS
1. Construct LH -derivations of the following logically valid sentences. To
make this easier, you may supplement the rules of LH with
A1 Ak
B
whenever B is a quasitautological consequence of A1 , . . . , Ak
Math 557 Midterm Exam #2
November 10, 2003
SOLUTIONS
1. Construct an LH -derivation of x (P x y P y ).
Solution.
1. P a y P y
EI
2. (P b P b) (P a y P y )
1, QT
3. (P b P b) x (P x y P y )
2, UG
4. x (P x y P y )
3, QT
2. Let A be the logically valid sent
Math 557
October 5, 2005
Homework #2 and Midterm #1
PARTIAL SOLUTIONS
1. Use signed tableaux to show that the following are logically valid.
(a) (x (A B ) (x A) (x B )
F (x (A B ) (x A) (x B )
T x(A B )
F (x A) (x B )
T x A
F x B
F B [x/a]
T A[x/a]
T (A B
Math 557 Homework #4
October 24, 2005
SOLUTIONS
1. Let L = cfw_R, . . . be a language which includes a binary predicate R.
Let S be a set of L-sentences. Assume that for each n 1 there
exists an L-structure (Un , Rn , . . .) satisfying S and containing el
Math 557 Final Exam
December 14, 2005
8 problems
1. Carefully state Vaughts Test, remembering to mention all of the required hypotheses.
2. Exhibit the axioms of the theory of random graphs.
3. Given positive integers n and k , let us say that n is good f