Question

If we let N stand for the set of all natural numbers, then we write 6N for the set of natural numbers all multiplied by 6 (so 6N = {6, 12, 18, 24, . . . }). Show that the sets N and 6N have the same cardinality by describing an explicit one-to-one correspondence between the two sets.

Answer #1

1.The one-to-one correspondence between the positive rational
numbers and the natural numbers implies what conclusion about the
cardinality of the two sets?
2. Is it possible to form a one-to-one correspondence between
the natural numbers and the real numbers? Is either set a proper
subset of the other? What is the significance of the answer to
these questions?

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

Using Discrete Math
Let ρ be the relation on the set of natural numbers N given by:
for all x, y ∈ N, xρy if and only if x + y is even. Show that ρ is
an equivalence relation and determine the equivalence classes.

Let N2K be the set of the first 2k natural numbers.
Prove that if we choose k + 1 numbers out of these 2k, there is at
least one pair of numbers a, b for which a is divisible by b.

Exercise 6.6. Let the inductive set be equal to all natural
numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 +
n.
(b) ∀n, 4n − 1 is divisible by 3.
(c) ∀n, 3n ≥ 1 + 2 n.
(d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.

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.

Show by induction that 1+3+5+...+(2n-1) = n^2 for all n in the
set of Natural Numbers

1. [10 marks] We begin with some mathematics regarding
uncountability. Let N = {0, 1, 2, 3, . . .} denote the set of
natural numbers.
(a) [5 marks] Prove that the set of binary numbers has the same
size as N by giving a bijection between the binary numbers and
N.
(b) [5 marks] Let B denote the set of all infinite sequences
over the English alphabet. Show that B is uncountable using a proof
by diagonalization.

Let S(n) be a
monotonic non-decreasing positive function deﬁned for all natural
numbers n. We do not know the value of S(n) for every n ∈ N except
when n = 2k for some k ∈ N, in which case S(n) = n log n
+ 3n − 5. Show that S(n) ∈ Θ(n log n).
Hint: (if you use it,
you need to prove it): ∀n > 1 ∈ N, ∃k ∈ N, such that
2k-1 ≤ n ≤...

Let S(n) be a
monotonic non-decreasing positive function deﬁned for all natural
numbers n. We do not know the value of S(n) for every n ∈ N except
when n = 2k for some k ∈ N, in which case S(n) = n log n
+ 3n − 5. Show that S(n) ∈ Θ(n log n).
Hint: (if you use it,
you need to prove it): ∀n > 1 ∈ N, ∃k ∈ N, such that
2k-1 ≤ n ≤...

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