CSE607: Exam-3 S-K Chin 2002 Theories with Equality Exam:
1
CSE607
Theories with Equality and Algebra Exam
Exam#3 April 1, 2002
This is an open-book and open-notes examination. Do all problems in 80 minutes. Make sure you justify your
steps using the formal inference rules to earn full credit. (This includes rewriting rules–this will help you to
avoid silly mistakes). Left justify assumptions and indent goals – it makes evaluating your proofs easier.
1. (15 points) Relations and Orders
(10 points for setting up the proof; 5 points for the proof)
Let R(x, y) be a relation that is
• Transitive
• Irreflexive
Let S(x, y) be a relation that is
• Transitive
• Symmetric
• Reflexive
Let relation Q be defined as
.. S(x,y)R(x, y)[Q(x, y)yx
∨
≡
∀
Prove there exists a case where Q(x, y) and Q(y, x)are both true.
2. (10 points) Prove the following
)](yz)xz[(y z y.x. G1.
z)(xyzx)(y A3.
x A2.
xex A1.
x
x
A0.
1
x
ex
=⊃=∀
=
=
=
=
−
3. (15 points) Prove the following
A1. Whoever cooperates with Deena will hire Cindy.
A2. Jenny hires only for friends of Laura.
A3. No one is friend of Kelly and has Cindy as a friend.
G1. If Laura is a friend of Kelly, Jenny will not cooperate with Deena.
A. (10 points of 15) Use the following notation to translate the above statements into symbolic
form:
• C(x, y): x cooperates with y
• H(x, y): x hires y
• F(x, y): x is a friend of y
• d: Deena, c: Cindy, j: Jenny, l: Laura, k: Kelly
B. (5 points of 15) Proven the goal G1 given assumptions A1, A2, and A3. Make sure you justify
each step in the tableau.
