Question

Real Topology: let A={1/n : n is natural} be a subset of the real
numbers. Is A open closed, or neither? Justify your answer.

Answer #1

For n in natural number, let A_n be the subset of all those real
numbers that are roots of some polynomial of degree n with rational
coefficients.
Prove: for every n in natural number, A_n is countable.

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....

Consider two copies of the real numbers: one with the usual
topology and one with
the left-hand-side topology. Endow the plane with the product
topology determined by these two
topologies.
Describe and give three examples of the open sets of the
topological subspace { (x,y)| y=x}.

Let n be a positive integer and let U be a finite subset of
Mn×n(C) which is closed under multiplication of matrices. Show that
there exists a matrix A in U satisfying tr(A) ∈ {1,...,n}

Let S(n) be the statement: The sum of the first n natural
numbers is 1/2 n2 + 1/2 n + 1000. Show that if S(k) is
true, so is S(k+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?

Using field axioms and order axioms prove the following
theorems
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions are given:
A subset S of R...

Let
U={1,2, 3, ...,3200}.
Let S be the subset of the numbers in U that are multiples of
4, and let T be the subset of U that are multiples of 9. Since
3200 divided by 4 equals it follows that
n(S)=n({4*1,4*2,...,4*800})=800
(a) Find n(T) using a method similar to the one that showed
that n(S)=800
(b) Find n(S∩T).
(c) Label the number of elements in each region of a two-loop
Venn diagram with the universe U and subsets S...

Let
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that
(a) There exist two elements of S that are relatively prime,
and
(b) There exist two elements of S, one of which divides the
other.

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.

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