Question

Question 3. Let a1,...,an ∈R. Prove that

(a_{1} + a_{2} + ... + a_{n})^{2}
/n ≤ (a_{1})^{2} + (a_{2})^{2} +
... + (a_{n})^{2}.

Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for
all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T).

Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above
and T is bounded below. Let U = {t−s|t ∈ T, s ∈ S}. Prove that U is
bounded below and inf(U) = inf(T)−sup(S).

Answer #1

Let n∈N, and let a1,a2,...an∈R. Prove that
|a1+a2+...+an|<or=|a1|+|a2|+...+|an|

(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪ ... ∪ An is countable. (Hint: Induction.)
(6) Let F be the set of all functions from R to R. Show that |F|
> 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)
(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4},
{1, 4}, {1, 2, 3, 4}}, and S =...

Let S and T be nonempty subsets of R with the following
property: s ≤ t for all s ∈ S and t ∈ T.
(a) Show that S is bounded above and T is bounded below.
(b) Prove supS ≤ inf T .
(c) Given an example of such sets S and T where S ∩ T is
nonempty.
(d) Give an example of sets S and T where supS = infT and S ∩T
is the empty set....

let A be a nonempty subset of R that is bounded below. Prove
that inf A = -sup{-a: a in A}

Please prove (a1+a2+……+an)^2/n ≤ (a1)^2 + (a2)^2 +……+
(an)^2.

Prove this statement: Let ϕ : A1 → A2 be a homomorphism and let
N = ker ϕ. Then A1/N is isomorphic to ϕ(A1). Further ψ : A1/N →
ϕ(A1) defined by ψ(aN) = ϕ(a) is an isomorphism.
You must use the following elements to prove:
- well-definedness
- one-to-one
- onto
- homomorphism

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

Consider the ring R = Z ∞ = {(a1, a2, a3, · · ·) : ai ∈ Z for
all i}. It turns out that R forms a ring under the operations (a1,
a2, a3, · · ·) + (b1, b2, b3, · · ·) = (a1 + b1, a2 + b2, a3 + b3,
· · ·), (a1, a2, a3, · · ·) · (b1, b2, b3, · · ·) = (a1 · b1, a2 ·
b2, a3 ·...

Let A⊆R be a nonempty set, which is bounded above. Let
B={a-5:a∈ A}. Prove that sup(B)=sup(A)-5

Let a1 = [
7
2
-1
]
a2 =[
-1
2
3
]
a3= [
6
4
9
]
a.)determine whether a1
a2 and a3span
R3
b.) is a3 in the Span {a1,
a2}?

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