Section 3.3
2)
a. True, Every non-empty set of
ℕ
has a min. Let S be a non-empty set of
ℕ
. If n
∈
S, then
S has a least element. Suppose n
∈
ℕ
for every m
∈
ℕ
, when m
≤
n-1. If m
∈
S, then
there is a least element in S.
b. False; Archimedean Property of
ℝ
states that if
ℕ
were bounded above then there must exist a
sup
ℕ
, called
a
such that
a
∈
ℕ
.
a
is the least upper bound of
ℕ
. Suppose there exist an n
∈
ℕ
such that n> a-1, where a-1 is not the sup
ℕ
. Then n+1 >
a
, and since n+1
∈
ℕ
. This
contradicts the statement that
a
is the sup
ℕ
. Therefore
a
cannot be the sup
ℕ
.
c. False, if x=y then xy=x*x=x
2
. Take x=
√
a
. Then xy=
√
a
2
¿
=a, which is rational.
d. True, Suppose that x>0. Then there exist an n
∈
ℕ
such that n > 1/(y-x)
ny-nx>1
Ny> 1+nx
Nx+1<ny
When nx
> 0.
e. False
f. False
3)
a. Sup=Max =3
b. Sup = Max =
π
c. Sup = Max = 4
d. Sup=4; No max
e. Sup=Max=1/2
f. Sup = Max = 0
g. Sup=1; No max
h. Sup= max= 3/2
i. No Sup; No max
j. Sup= 4; No max
k. Sup=max=1

l. Sup = 2; No Max
m. Sup = 5; No Max
n. Sup =

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- Fall '08
- Staff
- Empty set, Supremum, Order theory, sup, infimum