Question

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.

Answer #1

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.

15.)
a) Show that the real numbers between 0 and 1 have the same
cardinality as the real numbers between 0 and pi/2. (Hint: Find a
simple bijection from one set to the other.)
b) Show that the real numbers between 0 and pi/2 have the same
cardinality as all nonnegative real numbers. (Hint: What is a
function whose graph goes from 0 to positive infinity as x goes
from 0 to pi/2?)
c) Use parts a and b to...

Prove that the cardinality of of 2Z (the set of even integers)
is ℵ0.

for the function g(x) = 1/x for all nonzero real
numbers X. Is the cardinality the same for all even numbers as it
is for all integers

It is clear that the cardinality of the Natural numbers is no
more than the cardinality of the Rational numbers. Show that
Rational numbers have cardinality no greater than the natural
numbers (and therefore they have the same cardinality).

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.

Prove that [ 0, infinity) and (0, infinity) have the same
cardinality.

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

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.

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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