CSE 2315 
Discrete Structures
Homework 2: Predicate Calculus and Proof Techniques
CSE 2315 
Discrete Structures
Homework 2 Fall 2010
Due Date: Oct. 7 2010, 3:30 pm
Proofs using Predicate Logic
For all your predicate logic proofs you can use only the rules given in the following tables. In addition
you are allowed to apply the deduction method and to use the method of temporary hypotheses. All
other rules have to be proven ﬁrst.
Equivalence Rules
Rule Name
Expression
Equivalent Expression
Commutativity (comm)
P
∨
Q
Q
∨
P
P
∧
Q
Q
∧
P
Associativity (ass)
(
P
∨
Q
)
∨
R
P
∨
(
Q
∨
R
)
(
P
∧
Q
)
∧
R
P
∧
(
Q
∧
R
)
Distributivity (dis)
(
P
∨
Q
)
∧
R
(
P
∧
R
)
∨
(
Q
∧
R
)
(
P
∧
Q
)
∨
R
(
P
∨
R
)
∧
(
Q
∨
R
)
De Morgan’s Laws (De Morgan)
(
P
∨
Q
)
P
∧
Q
(
P
∧
Q
)
P
∨
Q
Implication (imp)
P
→
Q
P
∨
Q
Double negation (dn)
(
P
)
P
Selfreference (self)
P
∨
P
P
Negation (neg)
(
∀
x
)
P
(
x
)
(
∃
x
)
P
(
x
)
2010 Manfred Huber
Page 1
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Discrete Structures
Homework 2: Predicate Calculus and Proof Techniques
Inference Rules
Rule Name
From
Can Derive
Conjunction (con)
P
,
Q
P
∧
Q
Simpliﬁcation (sim)
P
∧
Q
P
,
Q
Modus ponens (mp)
P
,
P
→
Q
Q
Modus tollens (mt)
P
→
Q
,
Q
P
Addition (add)
P
P
∨
Q
Universal instantiation (ui)
(
∀
x
)
P
(
x
)
P
(
y
)
(
Be careful with the rule’s restrictions)
(
∀
x
)
P
(
x
)
P
(
a
)
Existential Instantiation (ei)
(
∃
x
)
P
(
x
)
P
(
y
)
(
Be careful with the rule’s restrictions)
(
∃
x
)
P
(
x
)
P
(
a
)
Universal generalization (ug)
P
(
x
)
(
∀
x
)
P
(
x
)
(
Be careful with the rule’s restrictions)
Existential generalization (eg)
P
(
x
)
(
∃
x
)
P
(
x
)
(
Be careful with the rule’s restrictions)
P
(
a
) (
∃
x
)
P
(
x
)
For all proofs the steps have to be annotated such as to indicate the rule and which elements of the
proof sequence it was applied to.
1. For each instantiation and generalization step in the following proof sequences indicate if it is
legal. You have to justify your decision (a short justiﬁcation is sufﬁcient).
a)
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 Spring '08
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 Natural number, Prime number, Manfred Huber, Discrete Structures Homework

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