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r
For each positive integer n we define P n to be the assertion that if a set has exactly n members
and if r is a nonnegative integer and r n then the set has exactly n subsets with r
r
members.
The assertion P 1 is obviously true. Now suppose that n is a positive integer for which the
assertion P n is true and suppose that S is a set with n 1 members, that r is a nonnegative
integer and that r n. 90 Choose a member of S that we shall call w.
Now given any subset E of S, if w E then E is the union of w and a subset of S
w that
contains r 1 members. Since the assertion P n is true, there are r n1 such subsets.
Furthermore, if E is a subset of S and E has r members and w does not belong to E then E is a
subset containing r members of the set S
w . Since the assertion P n is true there are n
r
such subsets.
The total number of subsets of S with n members is therefore
n
n
n1 .
r
r
r1
Since P 1 is true and since the condition P n P n1 holds for every positive integer n it follows
from the principle of mathematical induction that P n is true for every positive integer n.
10. Suppose that for every positive integer n we define p n to be the assertion that if in any crowd of men, at
least one of them has red hair then all of them have red hair. What is wrong with the following proof by
mathematical induction that the assertion p n is true for every n?
a. The statement p 1 is obviously true.
b. Now suppose that n is any positive integer for which the statement p n happens to be true and suppose
that S is a crowd containing n 1 men and that at least one of these n 1 men has red hair. Choose such
a man and let’s call him Harry. Now, in the crowd of n 1 men, choose any man and call him Joe and
ask him to step away from the others. There are n men left and, since at least one of them has red hair,
they all have. Now ask Harry to step away and ask Joe to come back. Again we are looking at a crowd of
n men and, since Joe is the only man about whom we have any doubt, at least one man in this crowd has
red hair; and therefore they all have red hair. Now ask Harry to come back and we see that all n 1 men
have red hair.
The argument given above has to refer to three different men. It refers to Harry and to Joe and
also to at least one other redhaired man. The argument therefore cannot be used to show that
P2 P3. Some Exercises on the Extended Real Number System
1. Thinking of the rule for sums of limits
lim f x g x
xa lim f x lim g x
xa
xa that you saw in elementary calculus, give some examples to show why the expression Ý
be defined. Solution: If we define f x x and g x 1 Ý should not x for every number x then lim f x Ý
xÝ
and
lim g x Ý
xÝ
lim f x g x
xÝ 1. With this example in mind we might conclude that if Ý Ý is to be defined, it ought to be equal to 1.
However, if we define f x x and g x 2 x for every x then we should conclude that Ý Ý ought
to be 2. If we define f x x sin x and g x x...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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