Lecture 10
Methods of Proof
CSCI – 1900 Mathematics for Computer Science
Fall 2014
Bill Pine
Lecture Introduction
• Reading
– Rosen - Section 1.7, 1.8
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•
•
•
•
•
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The Nature of Proofs
Components of a Proof
Rule of Inference and Tautology
Proving equivalences
modus ponens
Indirect Method
Proof by Contradiction
CSCI 1900
Lecture 10 - 2
Past Experience
• Until now we’ve used the following methods
to write proofs:
– Direct proofs with generic elements,
definitions, and given facts
– Proof by enumeration of cases such as when
we used truth tables
CSCI 1900
Lecture 10 - 3
General Outline of a Proof
• p  q denotes "q logically follows from p“
• Implication may take the form (p1  p2  p3
 …  pn)  q
• q logically follows from p1, p2, p3, …, pn
CSCI 1900
Lecture 10 - 4
General Outline (cont)
The process is generally written as:
p1
p2
p3
:
:
pn
 q
CSCI 1900
Lecture 10 - 5
Components of a Proof
• The pi's are called hypotheses or premises
• q is called the conclusion
• Proof shows that if all of the pi's are true,
then q has to be true
• If result is a tautology, then the implication
p  q represents a universally correct
method of reasoning and is called a rule of
inference
CSCI 1900
Lecture 10 - 6
Example of a Tautology based Proof
• If p implies q and q implies r, then p implies r
pq
qr
 pr
• By replacing the bar under q  r with the “”, the
proof above becomes ((p  q)  (q  r))  (p  r)
• The next slide shows that this is a tautology and
therefore is universally valid.
CSCI 1900
Lecture 10 - 7
Tautology Example (cont)
p q
r pq qr
T
T
T
T
F
F
T
F
T
F
T
F
T
T
F
F
T
T
T
F
T
T
T
F
F F T
F F F
T
T
T
T
T
T
F
F
T
T
(p  q) 
(q  r)
T
F
F
F
T
F
T
T
CSCI 1900
pr
T
F
T
F
T
T
T
T
((p  q)  (q  r))
 (p  r)
T
T
T
T
T
T
T
T
Lecture 10 - 8
Equivalences
• Some mathematical theorems are
equivalences, i.e., p  q.
– If and only if
– Necessary and sufficient
• The proof of such a theorem requires two
proofs:
– Must prove p  q, and
– Must prove q  p
CSCI 1900
Lecture 10 - 9
Rule of Inference: modus ponens
• modus ponens – the method of asserting
p
pq
 q
• Example:
– p: It is snowing today
– p  q: If it is snowing, there is no school
– q: There is no school today
• Supported by the tautology (p  (p  q))  q
CSCI 1900
Lecture 10 - 10
Show modus ponens = Rule of Inference
p
q
(p  q)
p  (p  q) (p  (p  q))  q
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
CSCI 1900
Lecture 10 - 11
Summary of Arguments
• An argument is a sequence of statements
– All statement except the last are premises
– The final statement is the conclusion
– Normally place the symbol  in front of the
conclusion
• To say that an argument is valid means only
that its form is valid
– If the premises are all true the conclusion is true
CSCI 1900
Lecture 10 - 12
Summary of Arguments (cont)
• It is possible for a valid argument to lead to
a false conclusion
• It is possible for an invalid argument to lead
to a true conclusion
• Carefully distinguish between validity of
the argument and truth of the conclusion
• A valid argument is one whose form, when
supplied with true premises cannot have a
false conclusion
CSCI 1900
Lecture 10 - 13
Invalid Conclusions from Invalid Premises
• Just because the format of the argument is valid does
not mean that the conclusion is true. A premise may
be false. For example:
Star Trek is real
If Star Trek is real we can travel faster than light
 We can travel faster than light
• Argument is valid since it is in modus ponens form
• Conclusion is false because a premise is false
CSCI 1900
Lecture 10 - 14
Invalid Conclusion from Wrong Form
• Sometimes, an argument that looks like modus ponens is
actually not in the correct form. For example:
If I am watching Star Trek, then I am happy
I am happy
 I am watching Star Trek
• Argument is not valid since its form is:
pq
q
 p
Which is not modus ponens
CSCI 1900
Lecture 10 - 15
Invalid Argument (continued)
• Truth table shows that this is not a tautology:
T
q (p  q) (p  q)  q ((p  q)  q) 
p
T
T
T
T
T
F
F
F
T
F
T
T
T
F
F
F
T
F
T
p
CSCI 1900
Lecture 10 - 16
Rule of Inference: Indirect Method
• Another method of proof is to use the tautology:
(p  q)  (~q  ~p)
• The form of the proof is (modus ponens):
~q
~q  ~p
~p
CSCI 1900
Lecture 10 - 17
Indirect Method Example
• p: My e-mail address is available on a web site
• q: I am getting spam
• p  q: If my e-mail address is available on a web
site, then I am getting spam
• ~q  ~p: If I am not getting spam, then my e-mail
address must not be available on a web site
• This proof says that if I am not getting spam, then
my e-mail address is not on a web site
CSCI 1900
Lecture 10 - 18
Another Indirect Method Example
• Prove that if the square of an integer is odd,
then the integer is odd too.
• p: n2 is odd
• q: n is odd
• ~q  ~p: If n is even, then n2 is even.
• If n is even, then there exists an integer m for
which n = 2×m. n2 therefore would equal
(2×m)2 = 4×m2 which must be even.
CSCI 1900
Lecture 10 - 19
Rule of Inference:modus tollens
• Another method of proof is to use the
tautology (p  q)  (~q)  (~p)
• The form of the proof is:
pq
~q
 ~p
• Also known as proof by contradiction
CSCI 1900
Lecture 10 - 20
Proof by Contradiction (cont)
~q
T T
(p 
q)
T
~p
F
(p  q) 
~q
F
F
(p  q)  (~q) 
(~p)
T
T F
F
T
F
F
T
F T
T
F
F
T
T
F F
T
T
T
T
T
p
q
CSCI 1900
Lecture 10 - 21
Proof by Contradiction (cont)
• The best application for this is where you
cannot possibly go through a large
(potentially infinite) number of cases to
prove that every one is true
CSCI 1900
Lecture 10 - 22
Key Concepts Summary
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•
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The Nature of Proofs
Components of a Proof
Rule of Inference and Tautology
Proving equivalences
modus ponens
Indirect Method
Proof by Contradiction
CSCI 1900
Lecture 10 - 23