Due 2007.Oct.05 (Fri) noon.
A hw04-makeup assignment will be posted later this weekend.
Although you are welcome to typeset your work nicely (using Microsoft Word
or LaTeX or whatever to get nice logic symbols),
it's probably much easier to write formulas by hand.
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Here's a screen shot.
Rosen p.132, #50
(= Rosen 5ed p. 96 #40):
bit-string representation.
00111 00000
10100 10001
01110 01110
Consider the function dorm,
which maps
an on-campus RU student to
their dorm-building
As expected,
the domain of dorm is on-campus RU students;
its codomain is RU dormitories.
For example, dorm(Jane Doe) = Muse.
Is this function onto?
Yes.
(Every building has some student staying in it.Okay, if occupancy
were horrendously low and there was an entire dorm with nobody in it,
then it wouldn't be onto. But we know that isn't currently the case.)
Is it one-to-one?
No. (It is many-to-one.)
What if the codomain were all campus buildings, instead of just dorms --
would the function still be one-to-one? onto?
Not be onto (since some buildings,
like the library, have no students who live there),
not one-to-one.
What if the codomain were individual beds in the dorm, instead of just entire dorms --
would the function still be one-to-one? onto?
Yes it would be one-to-one (assuming two students weren't assigned to the
same bed).
It would not be onto, since the occupancy isn't quite 100%.
Rosen p146, #2
(= Rosen 5ed p. 108 #2):
Is it a function from Z → ℜ?
(That is, if you put in an integer,
will you get out exactly one corresponding real number?)
f(n) = ±n.
Not a function -- given one input, there are two outputs.
But it is a function if the codomain is sets of numbers --
that is, f:N→P(N).
f(n) = √(n²+1).
This is a function.
(For every n, n² is positive, so n²+1 does have a real
(positiveNote that √9 = 3 (by definition of √),
even though there are two numbers which square to 9: both 3 and -3.) square root.)
f(n) = 1/(4-n²).
This is not quite a function:
it's not defined on the inputs 2 and -2.
Rosen p146, #8
(= Rosen 5ed p. 108 #8):
Ceiling, floor
1-13120-22
Rosen p146, #12
(= Rosen 5ed p. 108 #12):
one-to-one?
f(n)=n-1
yesf(n)=n²+1
nof(n)=n³
yesf(n)=ceiling(n/2)
no
For each of the functions in the previous problem,
is f : Z→Z onto?
yesnonoyes
If you see a few other problems in Rosen which catch your eye,
and you'd like to do them for extra credit, you are welcome to
(though you can ask me for how much; extra-credit is harder to earn
point-per-point than regular credit).
If you write your own html, you might be interested in this
page of useful html math (and other) entities