Homework #11
*********** Section 5.2 **********
3a)
false
3b)
true
3c)
false
3d)
true
3e)
true
3f)
false
4b) Define h:N>E+ by h(n)= { 16 if n=1, 12 if n=2, 2 if n=3, 4 if n=8,
6 if n=6, 2n else
5b) c
5d) Xo
5f) c
5g) Xo
6a) A=N, B=N{1}
6b) A=N, B=positive even integers
6c) A=N, B=N{1}
7a) A=(2,5), B=(3,4)
7b) A=R, B=R{0}
7c) A=(1,0), B=(3,5)
10.)
This is clearly a function on (0,1), as it is defined for all
numbers in this open interval.
If you compute the derivative you will
see that it is negative throughout the interval, the function is thus
decreasing, and it is thus 11.
As x gets arbitrarily close to 0 from the right, f gets arbitrarily
large.
As x gets arbitrarily close to 1 from the left, f gets
arbitrarily large negative.
We can then use the Intermediate Value
Theorem and the fact that f is continuous everywhere to conclude that f
takes on any real value.
f is thus onto.
11.)
[An intuitive note is that both RXR and C are represented
graphically with a coordinate system in the plane, so this suggests that
there is a 11 correspondence.
A convenient one would be to map (x,y)
to (x,y).]
To show this, let f:C > RXR be given by f(a+bi) = (a,b).
This is defined for all of C, and the assignments are unique, so it is a
function.
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 Winter '10
 Staff
 Set Theory, Natural number, Finite set, DENUMERABLE SETS

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