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Unformatted text preview: if we define j for each j
function from 1 to 1, 2, , m whose value at 1 is j then is a oneone function from 1, 2, , m onto
1 . Therefore
card 1 m m 1 .
Now suppose that n is any positive integer for which the assertion p n is true. In order to prove that
card n1 m n1 we shall show that
1, 2, m ß n1
n
and for this purpose we shall define a function
: n
1, 2, , m
n 1
as follows: For each member f, j of the set n
1, 2, , m we define
f, j k 1, 2, , n fk if k j if k n 1 . The proof will be complete when we have seen that is oneone and is onto the set n1 .
1, 2, , m and that
To see that is oneone we suppose that f 1 , j 1 and f 2 , j 2 belong to n
1, 2, , n we have
f 1 , j 1 f 2 , j 2 . For each k
f1 k f1, j1 k f2, j2 k f2 k
and so f 1 f 2 . We see also that
j1 f1, j1 n 1 f2, j2 n 1 j2.
Finally, to see that is onto the set n1 , suppose that g
n1 and define j g n 1 and define
f k g k for every k
1, 2, , n . We see at once that g f, j . 56 11. a. Prove that if for each positive integer n we define F n to be the set of all oneone functions from
1, 2, , n onto 1, 2, , n then for each n we have
F n 1 ß F n
1, 2, , n, n 1 . Solution: To motivate the solution of this exercise we should consider that a oneone function from
the set 1, 2, , n onto itself is a way or arranging the members of the set 1, 2, , n in a sequence
x 1 , , x n . For each such arrangement we can obtain an arrangement of the members of the larger set
1, 2, , n, n 1 by placing the number n 1 in any of the n 1 positions shown
n 1, x 1 , x 2 , , x n
x 1 , n 1, x 2 , , x n
x 1 , x 2 , , x n , n 1
Thus the number of ways of arranging the members of 1, 2, , n, n 1 should be n 1 times the
number of ways of arranging the members of 1, 2, , n in a sequence.
Now we begin: We define a function from F n
1, 2, , n, n 1
member f, m of the set F n
1, 2, , n, n 1 we define
fj if j m n1 if j m . fj f, m j F n1 as follows: Given any if j m 1 Each such function f, m is a oneone function from 1, 2, , n, n 1 onto 1, 2, , n, n 1 . To
see that the function is oneone, suppose that f, m and g, k belong to the set
1, 2, , n, n 1 and that f, m g, k . Since the function g, k is oneone and since
Fn
g, k m f, m m n 1 g, k k
we deduce that k m.
Given any j m we have
f j f, m j g, m j g j
and given any j m we have
f j f, m j 1 g, m j 1 g j
and we conclude that f g. Therefore the function is oneone.
Finally, to show that is onto the set F n1 we assume that h F n1 . In other words, h is a oneone
function from the set 1, 2, , n, n 1 onto 1, 2, , n, n 1 . We see easily that if m h 1 n 1
and if we define f : 1, 2, , n
1, 2, , n by the equation
hj then f if j m h j1 fj if j m F n and h f, m . b. Prove that for each positive integer n, the number of ways of ordering the numbers in the set 1, ...
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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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