CSE 20: Discrete Mathematics
Spring 2010
Problem Set 2
Instructor: Daniele Micciancio
Due on:
Thu. April 15, 2010
Problem 1 (6 points)
Formulate each of the following statements in English (without using mathematical symbols) and assert
which one is true or false:
1.
∀
x
∈
Z
.
∃
y
∈
Z
.x < y
For all integers there exist another integer which is larger than them.
≡
True.
2.
∃
y
∈
Z
.
∀
x
∈
Z
.x < y
There exist an integer which is larger than all integers.
≡
False.
Problem 2 (10 points)
Complete the proof of the following theorem by filling the gaps.
Theorem 1.
If
p
→ ¬
q
and
q
∨
r
, then
p
→
r
.
Proof.
Given
p
→ ¬
q
and
q
∨
r
, we need to prove
p
→
r
. In order to prove
p
→
r
, we assume
p
is true, and
show that
r
follows. We give a proof by cases. Since
q
∨
r
is given, either
q
or
r
is true. We consider the
following two cases:
•
Case 1 (
r
is true): In this case
r
is true and there is nothing to be proved.
•
Case 2 (
q
is true): Since
p
→ ¬
q
is given and
p
is true by assumption, we can deduce
¬
q
. But we also
have
q
. This proves
q
∧ ¬
q
which is a contradiction. So, also in this case
r
follows.
In both cases we proved that
r
is true, and this concludes the proof of the theorem.
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 Fall '08
 Foster
 Alice, Natural deduction, ak b

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