Download Worksheet I: What is a proof (And what is not a proof)

Worksheet I: What is a proof (And what is not a
proof)
1. Complete the following “proof without words” that appears in
Nelson’s Proofs Without Words.
2.
For each of the two Abbott & Costello videos and the Colbert video (cf.
References), determine if it can be expressed as a sequence of logical statements or
not. If you think it cannot, then explain.
3. Express as a logical proof (Russell’s statement): “If -1 = 1 then I am the Pope.”
4. Proof or not?
1  1  (1)(1)   1  1 
  1
2
 1
from: Michael Stueben and Diane Sandford, Twenty Years Before the Blackboard, MAA (1998)
5. Proof or not?
53
5 log 10 (1 / 3)  3 log 10 (1 / 3)
log 10 (1 / 3) 5  log 10 (1 / 3) 3
(1 / 3) 5  (1 / 3) 3
1 / 243  1 / 27
6.
Proof or not?
1 ¢ =$0.01 = ($0.1)2 = (10¢)2 = 100¢ = $1.00
7.
(A. Meyer, 6.042J)
Proof or not?
Let a and b be two equal real numbers. Then
a=b
a2 = ab
a2 – b2 = ab – b2
(a – b)(a +b) = (a – b)b
a+b=b
a=0
8. (A. Meyer, 6.042J)
It is a fact that the arithmetic mean is greater than or equal to the geometric mean
for all non-negative real numbers, a and b.
ab
 ab
That is:
Is the following a proof of this statement or not?
2
ab ?
 ab
2
a  b ? 2 ab
a 2  2ab  b 2 ? 4ab
a 2  2ab  b 2 ? 0
( a  b) 2  0
which we know to be true
9. (A. Meyer, 6.042J)
10. Unexpected Hanging Paradox
A paradox also known as the surprise examination paradox or prediction paradox.
A prisoner is told that he will be hanged on some day between Monday and Friday, but
that he will not know on which day the hanging will occur before it happens. He cannot
be hanged on Friday, because if he were still alive on Thursday, he would know that the
hanging will occur on Friday, but he has been told he will not know the day of his
hanging in advance. He cannot be hanged Thursday for the same reason, and the same
argument shows that he cannot be hanged on any other day. Nevertheless, the executioner
unexpectedly arrives on some day other than Friday, surprising the prisoner.
This paradox is similar to that in Robert Louis Stevenson's "bottle imp paradox," in
which you are offered the opportunity to buy, for whatever price you wish, a bottle
containing a genie who will fulfill your every desire. The only catch is that the bottle
must thereafter be resold for a price smaller than what you paid for it, or you will be
condemned to live out the rest of your days in excruciating torment. Obviously, no one
would buy the bottle for 1¢ since he would have to give the bottle away, but no one
would accept the bottle knowing he would be unable to get rid of it. Similarly, no one
would buy it for 2¢, and so on. However, for some reasonably large amount, it will
always be possible to find a next buyer, so the bottle will be bought (Paulos 1995).
11. Russell's paradox (1903) is based on examples such as this: Consider a group of
barbers who shave only those men who do not shave themselves. Suppose there is
a barber in this collection who does not shave himself; then by the definition of the
collection, he must shave himself. But no barber in the collection can shave
himself. (If so, he would be a man who does shave men who shave themselves.)
12. Here’s the basic Berry paradox, by the way, if you might want an example of why
logicians with incredible firepower can devote their whole lives to solving these
things and still end up beating their heads against the wall. This one has to do with
big numbers — meaning really big, past a trillion, past ten to the trillion to the
trillion, way up there. When you get way up there, it takes a while even to describe
numbers this big in words. ‘The quantity one trillion, four hundred and three
billion to the trillionth power’ takes twenty syllables to describe, for example. You
get the idea. Now, even higher up there in these huge, cosmic-scale numbers,
imagine now the very smallest number that can’t be described in under twenty-two
syllables. The paradox is that the very smallest number that can’t be described in
under twenty-two syllables, which of course is itself a description of this number,
only has twenty-one syllables in it, which of course is under twenty-two syllables.
So now what are you supposed to do?
- David Foster Wallace, Good old neon, Oblivion
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