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CS103
HO#7
Proof Methods
4/4/11
1
Suppose we can prove that q follows from p.
There are two ways to think about the proof:
p
s
1
...
s
n
q
Suppose p is true
s
1
...
s
n
q
p
q
Doing Real Proofs: Example
Prove:
If n is even, then n
2
is even.
That is,
Prove: Even(n)
Even(n
2
)
Problem: the formula above is not a sentence in FOL, since
n appears as a free variable.
This is better:
x (Even(x)
Even(x
2
))
but what is the domain of the quantifier?
Doing Real Proofs: Example
Prove:
If n is even, then n
2
is even.
That is,
Prove: Even(n)
Even(n
2
)
Problem: the formula above is not a sentence in FOL, since
n appears as a free variable.
What we really mean:
Prove:
x (Even(x)
Even(x
2
)), where the domain is integers
So now we need rules of inference for quantifiers.
x (P(x)
Q(x))
z (Q(z)
R(z))
x(P(x)
R(x))
Can we just use the rules we have inside the quantifiers?
x (P(x)
Q(x))
z (Q(z)
R(z))
x(P(x)
R(x))
No!
Example: Statementreason proof
Prove:
For all integers, if n is even, then n
2
is even.
1.
Let c be an
arbitrary
integer, and
suppose Even(c).
2.
k (c = 2k)
Def. of Even, 1
3.
c
2
= 4k
2
Algebra, 2
4.
c
2
= 2m, where m = 2k
2
Algebra, 3
5.
Even(c
2
)
Def. of Even, 4
6.
x (Even(x)
Even(x
2
))
General Conditional
Proof, 1  5
Example: Statementreason proof
Prove:
For all integers, if n is even, then n
2
is even.
1.
Let c be an
arbitrary
integer, and
suppose Even(c).
2.
k (c = 2k)
Def. of Even, 1
3.
c
2
= 4k
2
Algebra, 2
4.
c
2
= 2m, where m = 2k
2
Algebra, 3
5.
Even(c
2
)
Def. of Even, 4
6.
x (Even(x)
Even(x
2
))
General Conditional
Proof, 1  5
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View Full DocumentCS103
HO#7
Proof Methods
4/4/11
2
Example: Paragraph form
Prove:
For all integers, if n is even, then n
2
is even.
PROOF:
Let c be an arbitrary integer, and suppose c is even.
We will show that c
2
is even.
Since c is even, there exists an integer k such that c = 2k
by the definition of even.
Thus c
2
= 4k
2
, and c
2
= 2m where m = 2k
2
, which means
that c
2
is even.
■
Rules of Inference for Quantified Statements
x P(x)
P(c)
where c is an object in the domain
Universal Instantiation
P(c)
where c is an object in the domain
x P(x)
Existential Generalization
Rules of Inference for Quantified Statements
x P(x)
P(c)
for some object c in the domain,
where c is a new name in the proof
Existential Instantiation
...
x P(x)
Give the name c to such an object
P(c)
...
Rules of Inference for Quantified Statements
P(c)
where c is an arbitrary object in the domain
x P(x)
Universal Generalization
Let c be an arbitrary object in the domain
...
P(c)
x P(x)
Every child is righthanded or intelligent
No intelligent child eats liver
There is a child who eats liver and onions
There is a righthanded child who eats onions
1.
x (R(x)
I(x))
All children are righthanded or intelligent
2.
x (I(x)
¬L(x))
No intelligent child eats liver
3.
x (L(x)
O(x))
Some child eats liver and onions
4.
L(a)
O(a)
For some a, Existential Inst., 3
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