CSE 331: Introduction to Algorithm Analysis and Design
Fall 2010
Homework 0 Solutions
Do not turn in
This homework is to refresh your memory on stuﬀ you should have seen in CSE 191 and/or CSE
250. This homework is intended for you to start thinking about proofs again.
Do not turn this
one in, it will NOT be graded.
1. (
What is a proof?
) (40
points
) Consider the following “proof”:
•
Brad Pitt has a beard
•
Every goat has a beard
•
Hence, Brad Pitt is a goat.
State precisely where the “proof” above fails logically.
Followup question: Can you prove that Brad Pitt is not a goat?
Proof Idea
:
The main ﬂaw in the proof above is that the conclusion seems to require the
following statement as the second statement
Everyone who has a beard is a goat.
Also there is insuﬃcient information to
prove
that Brad Pitt is not a goat, even though we
know from “common knowledge” that Brad Pitt is not a goat. However, in problems that
you solve in this course you
cannot
assume anything outside of what is given in the problem
statement.
More formally, we’ll use predicate logic to show the ﬂaw in the proof that
Brad Pitt
is a
goat. First, we’ll construct a few predicates which we can use to express the given statements.
Let
b
(
x
) represent the statement “x has a beard”. Let
g
(
x
) represent the statement “x is a
goat”.
2
Solution
: The given proof, in our predicate notation, is:
b
(
Brad Pitt
)
∀
x
(
g
(
x
)
→
b
(
x
))
Therefore
g
(
Brad Pitt
)
In order to draw the conclusion
g
(
Brad Pitt
) from
b
(
Brad Pitt
), we would need the
conditional statement
∀
x
(
b
(
x
)
→
g
(
x
)). However, we don’t know that to be true, and it’s
not equivalent to any of our statements, so this “proof” fails. We can, however, use a state
ment that is logically equivalent to the conditional statement we do have,
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