1. Introduction to Proofs

2. Introduction
A proof is a valid argument establishing the truth of a mathematical statement.
We (CS) are interested in proof to establish
the correctness of an algorithm
the correctness of a program
a property of a system
(e.g., Java can be parsed efficiently)
AI, automated theorem proving
the consistency of a specification
. . .

3. Methods for Proving Theorems
The theorem typically is of the form
x, ( P(x)  Q(x) ).
The proof method typically is of the form
Let c be an arbitrary c in the domain..
Show that P(c)  Q(c) is true. (The essence.)
Therefore, x, ( P(x)  Q(x) ).

4. Direct Proof of p  q Assume that p is true. Show that q is true.
Let the integers, Z, be the domain.
Prove that x (( 3 | x – 2 )  ( 3 | x2 – 1 ))
3 | x – 2 (Assumption)
x – 2  0 modulo 3.
x  2 modulo 3.
4. x2  1 mod 3.
x2 – 1  0 mod 3.
3 | x2 – 1.

5. Proof by Contraposition of p  q
Direct proof of the contrapositive:
Assume ~q is true.
Show that ~p is true.

6. Proof by Contraposition of p  q
Example
Let the domain be the integers.
Prove: (ab is even)  (a is even  b is even)
Assume (a is even  b is even).
a is odd  b is odd.
m, a = 2m + 1; n, b = 2n + 1.
ab = (2m + 1)(2n + 1)
= 4mn + 2m + 2n + 1
= 2(2mn + m + n) + 1 is odd.

7. Proof by Contradiction of p  q
Show p  q is true by showing that
(p  q) ≡ ( p  q) ≡ (p  q) is false.

8. Example
If , 2, …, 10 are placed randomly in a circle then the sum of some 3 adjacent numbers  17.
1. Assume that 1, 2, …, 10 are placed randomly in a circle and that the sum of no 3 adjacent numbers  17.
2. Then x1 + x2 + x3  16
x2 + x3 + x4  16 …
x10 + x1 + x2  16.
3. Summing, 3( x1 + x x10 )  16  10 = 160.
4. But, 3 (x1 + x x10 ) = 3( … + 10)
= 3  55 = 165, a contradiction.

9. Proving p  q p  q means p if and only if q p if q means q  p
p only if q means if q then p, which means p  q.
So, p  q means p  q  q  p.
To prove p  q it thus is necessary and sufficient to show that p  q  q  p.

10. Characters
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       