Question

Define the 'closure' S of a set S of real numbers.

State as many equivalent characterizations, or results, about the closure S.

Answer #1

I only state the results, because in question it is wanted only the statements.

Prove: Let S be a bounded set of real numbers and let a > 0.
Define aS = {as : s ∈ S}. Show that inf(aS) = a*inf(S).

If a,b are elements of R(set of real numbers) and a<b, show
that [a,b] is equivalent to [0,1].

Let S be the set of real numbers between 0 and 1, inclusive;
i.e. S = [0, 1]. Let T be the set of real numbers between 1 and 3
inclusive (i.e. T = [1, 3]). Show that S and T have the same
cardinality.

Let S be the collection of all sequences of real numbers and
define a relation on S by {xn} ∼ {yn} if and only if {xn − yn}
converges to 0.
a) Prove that ∼ is an equivalence relation on S.
b) What happens if ∼ is defined by {xn} ∼ {yn} if and only if
{xn + yn} converges to 0?

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of nonnegative real numbers.

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of non-negative real numbers.

1. (a) Let S be a nonempty set of real numbers that is bounded
above. Prove that if u and v are both least upper bounds of S, then
u = v.
(b) Let a > 0 be a real number. Deﬁne S := {1 − a n : n ∈ N}.
Prove that if epsilon > 0, then there is an element x ∈ S such
that x > 1−epsilon.

Find a bijection between set of infinite subsets of natural
numbers and real numbers.
Find a bijection between set of finite subsets of real numbers
and real numbers.
Find a bijection between set of countable subsets of real
numbers and real numbers.

The following are attempts to define a binary operation on a
set, are they actually binary operations
on the given set? If yes, prove it and if not please provide an
explanation.
1) a*b = a-b on S, S is the set Z of integers.
2) a*b = a log b on S, S is the set R+ of positive real
numbers
3) a*b = |a+b| on S, S is the set of Real numbers.
what I want to know...

The Principle of Mathematical Induction is:
A: A set of real numbers
B: A set of rational numbers
C: A set of positive integers
D: A set of negative integers

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