Question

Let S be a nonempty set in R^{n}, and its support
function be

σ_{S} = sup{ <x,z> : z ∈ S}.

let conv(S) denote the convex hull of S. Show that σ_{S}
(x)= σ_{conv(S)} (x), for all x ∈ R^{n}

Answer #1

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 A be a nonempty set. Prove that the set S(A) = {f : A → A |
f is one-to-one and onto } is a group under the operation of
function composition.

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 ⋆ be an operation on a nonempty set S. If S1, S2 ⊂ S are
closed with respect to ⋆, is S1 ∪ S2 closed with respect to ⋆?
Justify your answer.

Real Analysis I
Prove the following exercises (show all your work)-
Exercise 1.1.1: Prove part (iii) of Proposition
1.1.8. That is, let F be an ordered field and x, y,z ∈ F. Prove If
x < 0 and y < z, then xy > xz.
Let F be an ordered field and x, y,z,w ∈ F. Then:
If x < 0 and y < z, then xy > xz.
Exercise 1.1.5: Let S be an ordered set. Let A
⊂...

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.

For a nonempty subset S of a vector space V , define span(S) as
the set of all linear combinations of vectors in S.
(a) Prove that span(S) is a subspace of V .
(b) Prove that span(S) is the intersection of all subspaces that
contain S, and con- clude that span(S) is the smallest subspace
containing S. Hint: let W be the intersection of all subspaces
containing S and show W = span(S).
(c) What is the smallest subspace...

Prove Cantor’s original result: for any nonempty set (whether
finite or infinite), the cardinality of S is strictly less than
that of its power set 2S . First show that there is a one-to-one
(but not necessarily onto) map g from S to its power set. Next
assume that there is a one-to-one and onto function f and show that
this assumption leads to a contradiction by defining a new subset
of S that cannot possibly be the image of...

Discrete mathematics function relation
problem
Let P ∗ (N) be the set of all nonempty subsets of N. Define m :
P ∗ (N) → N by m(A) = the smallest member of A. So for example, m
{3, 5, 10} = 3 and m {n | n is prime } = 2.
(a) Prove that m is not one-to-one.
(b) Prove that m is onto.

Let A be a nonempty set and let P(x) and Q(x) be open
statements. Consider the two statements (i) ∀x ∈ A, [P(x)∨Q(x)] and
(ii) [∀x ∈ A, P(x)]∨[∀x ∈ A, Q(x)]. Argue whether (i) and (ii) are
(logically) equivalent or not. (Can you explain your answer
mathematically and by giving examples in plain language ? In the
latter, for example, A = {all the CU students}, P(x) : x has last
name starting with a, b, ..., or h,...

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