Massachusetts Institute of Technology
Course Notes 1±
6.042J/18.062J, Fall ’02
: Mathematics for Computer Science±
Professor Albert Meyer
and
Dr. Radhika Nagpal±
Proofs±
1
What is a Proof?
A proof is a method of ascertaining truth. There are many ways to do this:
Jury Trial
Truth is ascertained by twelve people selected at random.
Word of God
Truth is ascertained by communication with God, perhaps via a third party.
Word of Boss
Truth is ascertained from someone with whom it is unwise to disagree.
Experimental Science
The truth is guessed and the hypothesis is confirmed or refuted by experi
ments.
Sampling
The truth is obtained by statistical analysis of many bits of evidence. For example,
public opinion is obtained by polling only a representative sample.
Inner Conviction/Mysticism
“
My
program is perfect. I know this to be true.”
“I don’t see why not...”
Claim something is true and then shift the burden of proof to anyone
who disagrees with you.
“Cogito ergo sum”
Proof by reasoning about undefined terms.
This Latin quote translates as “I think, therefore I am.” It comes from the beginning of a
famous essay by the 17th century Mathematician/Philospher, Rene
´ Descartes. It may be one
of the most famous quotes in the world: do a web search on the phrase and you will be
ﬂooded with hits.
Deducing your existence from the fact that you’re thinking about your existence sounds like
a pretty cool starting axiom. But it ain’t Math. In fact, Descartes
goes on
shortly to conclude
that there is an infinitely beneficent God.
Mathematics also has a specific notion of “proof” or way of ascertaining truth.
Definition.
A
formal proof
of a
proposition
is a chain of
logical deductions
leading to the proposition
from a base set of
axioms
.
The three key ideas in this definition are highlighted: proposition, logical deduction, and axiom.
Each of these terms is discussed in a section below.
Copyright © 2002, Prof. Albert R. Meyer.
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2
Course Notes 1: Proofs
2
Propositions
Definition.
A
proposition
is a statement that is either true or false.
This definition sounds very general, but it does exclude sentences such as, “Wherefore art thou
Romeo?” and “Give me an A!”.
Proposition 2.1.
2 + 3 = 5.
This proposition is true.
Proposition 2.2.
Let
p
(
n
)
::=
n
2
+
n
+
41
.
∀
n
∈
N
p
(
n
)
is a prime number
.
The symbol
∀
is read “for all”. The symbol
N
stands for the set of
natural numbers
, which are 0, 1,
2, 3,
. . .
; (ask your TA for the complete list). A
prime
is a natural number greater than one that is
not divisible by any other natural number other than 1 and itself, for example, 2, 3, 5, 7, 11,
. . .
.
Let’s try some numerical experimentation to check this proposition:
p
(0)
=
41
which is prime.
p
(1)
=
43
which is prime.
p
(2)
=
47
which is prime.
p
(3)
=
53
which is prime.
. . .
p
(20)
=
461
which is prime. Hmmm, starts to look like a plausible claim. In fact we can keep checking through
n
=
39
and confirm that
p
(39)
=
1601
is prime.
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 Spring '05
 HUANG
 Computer Science, Logic, Prime number, Proposition

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