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Recitation 3 Warmup SolutionsBrought to you by your sleepy
TA.
44. Show that if x is a real number and n is an integer, thena.
x n if and only if x n
The if direction is trivial, since x x . For the other
direction,suppose that x n . Since n is an integer no smaller than
x, and x is bydefinition the smallest such integer, clearly x n
.
b. n x if and only if n x
The if direction is trivial, since x x . For the other
direction,
suppose that n x . Since n is an integer not exceeding x, and x
is bydefinition the largest such integer, clearly n x .
53. Data are transmitted over a particular Ethernet network in
blocks of 1500 octets(blocks of 8 bits). How many blocks are
required to transmit the followingamounts of data over this
Ethernet network? (Note that a byte is a synonym for an octet, a
kilobyte is 1000 bytes, and a megabyte is 1,000,000 bytes. Of
course,we know thats not true, but we can pretend.)
a. ( )1
150,000 1001,500
block octets blocks
octets
=
b. ( )1
384,000 2561,500
block octets blocks
octets
=
c. ( )1
1,544,000 10301,500
block octets blocks
octets
=
d. ( )1
45,300,000 30, 2001,500
block octets blocks
octets
=
58. Draw the graph of the function f (x) = x + 2 x from R to R
.
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Recitation 3 Warmup SolutionsBrought to you by your sleepy
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64. Suppose that f is a function from A to B, where A and B are
finite sets with |A| = |B|. Show that f is one-to-one if and only
if it is onto.
If f is one-to-one, then every element of A gets sent to a
different element of B. If in addition to the range of A there were
another element in B, then | B| would be atleast one greater than |
A|. In particular, there is an element of A for each elementof B.
If two or more elements of A were sent to the same element of B,
then | A|would be at least one greater than the | B|. This cannot
happen, so we concludethat f is one-to-one.
69. See back of your textbook.
70. a) Show that a partial function from A to B can be viewed as
a function f * from A to { } B u where u is not an element of B
and
( )
*( )
f a
f a
u=
We merely have to remark that f * is well-defined by the rule
given here.For each a A , either a is in the domain of the
definition of f or it is not.If it is, then f *(a ) is the
well-definied element ( ) f a B , and otherwise
f *(a) = u. In either case f *(a ) is a well-defined element of
{ } B u .
b) Using the construction in (a), find the function f *
corresponding to each partialfunction in Exercise 69.
We simply need to set f *(a ) = u for each a not in the domain
of definitionof f . In part (a), then, f *(n) = 1/n for n 0, and f
*(0) = u. In part (b) we
have a total function already, so f *(n) =2n
for all n Z . In part (c)
f *(m,n) = m/n if n 0, and f *(m, 0) = u for all m Z . In part
(d) we havea total function already, so f *(m,n) = mn for all
values of m and n. In part(e) the rule only applies if m > n, so
f *(m,n) = m n if m > n, and f *(m,n)
= u if m < n.
48. Let f be a one-to-one function from the set A to the set B.
Let S and T be subsetsof A. Show that ( ) ( ) ( ) f S T f S f T =
.
if a belongs to the domain of definition of f
if f is undefined at a
