Name __________________________________________
MAT101 – Survey of Mathematical Reasoning (Honors)
Professor Pestieau
Spring 2011
Take-Home Exam 4 – Arguments & Mathematical Induction
Due on Tuesday, May 17
Present all your work, neatly, on the following problems to receive full credit.
ARGUMENTS
Problem 1
[5 pts each]
For the 6 arguments presented below, proceed as follows:
1) Break down the argument
Explicitly identify the component statements that are part of the argument and show
the symbolic form of the argument.
2) Analysis
Decide whether the argument is valid or invalid. Here you may use truth tables,
standard forms, Euler diagrams, and/or logical manipulation. If the argument is
invalid, point to its logical fallacy.
1.
“If we evolved a race of pure logicians, that would not be progress; but we have not
evolved a race of pure logicians; so that’s progress.”
2.
“Homer is a visionary or he’s a fool; but he’s certainly a visionary; so he’s not a fool.”
3.
“If the fossil is an ammonite, then it is from the Crestaceous period. If the fossil is not
from the Mesozoic era, then it is not from the Crestaceous period. If the fossil is from
the Mesozoic era, then it is at least 65 million years old. Therefore, if the fossil is an
ammonite, it is at least 65 million years old.”
4.
“No odd number has a factor of 2. Some prime numbers are odd. Therefore, some
prime numbers have a factor of 2.”
5.
This absurd argument is taken from Lewis Carroll’s Symbolic Logic (1896):
“Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons
are despised. Hence, babies cannot manage crocodiles.”
6.
“All Italian villas are wonderful. It is not wise to invest in expensive villas. Some
wonderful villas are expensive. Therefore, it is not wise to invest in Italian villas.”
Problem 2
[5 pts]
Determine, by any means, whether the following argument is valid or not:
“If a set A is a proper subset of another set B, then the intersection of A and B is also a
proper subset of B; but A is equal to B or B is a proper subset of A; hence the
intersection of A and B is a proper subset of A.”
Problem 3
[5 pts]
Find a valid conclusion to the argument given below in symbolic form. Justify your answer.
~ ( p ~ q )
~ r ~ q
~s
~ rs
---------------?
(premise 1)
(premise 2)
(premise 3)
(premise 4)
(conclusion)
Problem 4
[10 pts]
Consider an argument whose symbolic form is given by
p or q
~q
---------------- ,
 p
Where p and q are any two component statements.
a)
Show that this argument is valid using a circuit.
b)
If the first premise is changed to the exclusive disjunctive form “Either p or q ,” does the
argument remain valid? Justify your answer.
MATHEMATICAL INDUCTION
Problem 5
[10 pts]
Show that the following statements are true for all natural numbers n :
  2n  1  n2
1)
1 3  5 
2)
n 2  n  2 is even.
3)
A set with n elements has 2 n distinct subsets.
[Bonus – 5pts]
Bonus Problems
[5 pts each]
Both problems below are taken from Lewis Carroll’s celebrated tale Alice in Wonderland (1865).
1.
In Chapter 6, Alice has a conversation with the Cheshire Cat, who assures her that they
are both mad. She does not accept his argument that she is mad ("You must be, or you
wouldn't have come here"), but asks instead how he is sure of his own madness:
"To begin with," said the Cat, "a dog's not mad. You grant that?"
"I suppose so," said Alice.
"Well, then," the Cat went on, "you see a dog growls when it's angry, and wags its tail
when it's pleased. Now I growl when I'm pleased, and wag my tail when I'm angry.
Therefore I'm mad." [Chapter VI]
In his argument, the Cat commits a logical fallacy.
argument symbolically.
2.
Show it by breaking down his
“You should say what you mean,' the March Hare went on.
'I do,' Alice hastily replied; 'at least — at least I mean what I say — that's the same
thing, you know.'
'Not the same thing a bit!' said the Hatter. 'You might just as well say that "I see what I
eat" is the same thing as "I eat what I see"!'
'You might just as well say,' added the March Hare, 'that "I like what I get" is the same
thing as "I get what I like"!'
'You might just as well say,' added the Dormouse, who seemed to be talking in his
sleep, 'that "I breathe when I sleep" is the same thing as "I sleep when I breathe"!'"
[Chapter VII]
In the excerpt above, what is the logical fallacy in Alice’s argument? Show it by breaking
down her argument symbolically.