MATH 187 Quiz 8 Key
24.13. We know
sin : R R
which is neither one-to-one or onto. We note that the image of sin is the interval
[1, 1]. If we now write
sin : R [1, 1] ,
then we now have that sin is onto the interval [1, 1]. If we restrict the domain
to th
MATH 187 Quiz 7 Key
1) Prove
1 3 + 2 3 + 3 3 + + n3 =
2
n (n + 1)
2
(1)
for all positive integers n.
a) Proof by smallest counterexample. So assume otherwise. That is, we assume there is a smallest positive integer x such that (1) is false for n = x.
Thus
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MATH 187 Quiz 9 Key
Exercise 26.9
Let A, B , and C , be sets such that f : A B and g : B C .
a) Prove that if f and g are one-to-one, then so is g f .
Proof: Let x, y A and let (g f ) (x) = (g f ) (y ). We will be done when
we show that x = y. Since (g f