most recent semester
lect04c
proofs; sets
We saw:
direct proof (of a for-all).
Direct proof (of a for-exists):
It's easy *if* you can find a witness.
e.g. Prove that some perfect square is the sum
of two other perfect squares.
If you realize that 25 satisfies this (since 25 = 16 + 9).
What about:
e.g. Prove that some perfect square is the sum
of three other perfect squares?
(Hint: do zero arithmetic.)
Finally: how about "three positive perfect squares"?
... must search. 1? 4? 9? (Aha, 9!)
Strategies
- Use your definitions, to go a step or two forwards.
- Write down your goal. Use def'ns, to go a step or two backwards.
(This often helps you solidifies the meaning of the definitions for you.)
- Try a direct proof, if you can come up with a constructive algorithm;
try an indirect proof if you are trying to show how you think *any*
constructive algorithm must fail.
(This is the same thought process used in designing an algorithm,
and in debugging:
Figure out how your algorithm will (eg) search through *any* list-of-names.
To debug, come up with the smallest example which the algorithm won't work for.
- Another strategy: see if you can show the contrapositive.
Example: If m^2 is even, then m is even.
- To show "a iff b":
- I. show "if a, then b"
- II. show "if b, then a"
Example:
A number is rational iff its decimal expression repeats.
Similarly, "TFAE" ("The following are equivalent"):
- x is irrational
- the decimal expansion of x never repeats
- 3x is irrational.
Proof by cases:
Given a balance scale and 9 coins (8 real, and one light counterfeit),
you can find the counterfeit in 2 weighings.
http://en.wikipedia.org/wiki/Four_color_theorem
wikipedia 4-coloring, 1500 cases:
"There has to be a state with 5 or fewer neighbors;
If that state has 3 neighbors, then ...
If that state has 4 neighbors, then ...
If that state has 5 neighbors, then...
a mere 1936 'configurations'."
(Later reduced to 1476)
Proof by contradiction:
There are a finite number of primes.
Chomp example, from book.
most recent semester
rad�ford.edu