Question
In: Advanced Math
1: Let X be the set of all ordered triples of 0’s and 1’s. Show that...
1: Let X be the set of all ordered triples of 0’s and 1’s. Show that X consists of 8 elements and that a metric d on X can be defined by ∀x,y ∈ X: d(x,y) := Number of places where x and y have different entries.
2: Show that the non-negativity of a metric can be deduced from only Axioms (M2), (M3), and (M4).
3: Let (X,d) be a metric space. Show that another metric D on X can be defined by ∀x,y ∈ X: D(x,y) := d(x,y)/(1 + d(x,y)).
4: Let (X,d) be a metric space.
- Show that every open d-ball is a d-open subset of X.
- Show that every closed d-ball is a d-closed subset of X.
5: Let (X,d) be a metric space. Show that a subset A of X is d-open if and only if it is the union of a (possibly empty) set of open d-balls.
Solutions
Expert Solution
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You have asked 5 questions. I have answered the first one.
Colby Messinger answered 2 years agoRelated Solutions
Let V be the set of all ordered triples of real numbers. For u = (u1,...
Let V be the set of all ordered triples of real numbers. For u =
(u1, u2, u3) and v = (v1, v2, v3), we define the following
operations of addition and scalar multiplication on V :
u + v = (u1 + v1, u2 + v2 − 1, u3 + v3 − 2) and ku = (ku1, ku2,
ku3).
For example, if u = (1, 0, 3), v = (2, 1, 1), and k = 2 then
u +...
Let S be a set of n numbers. Let X be the set of all subsets...
Let S be a set of n numbers. Let X be
the set of all subsets of S of size k, and let
Y be the set of all ordered k-tuples
(s1, s2, ,
sk)
such that
s1 < s2
< < sk.
That is,
X
=
{{s1, s2, ,
sk} | si S and all si's
are distinct}, and
Y
=
{(s1, s2, ,
sk) | si S and s1 <
s2 < < sk}.
(a) Define a one-to-one correspondence
f : X → Y.
Explain...
Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and...
Let S be the set of all ordered pairs of real numbers. Define
scalar multiplication and addition on S by:
α(x1,x2)=(αx1,αx2)
(x1,x2)⊕(y1,y2)=(x1 +y1,0)
We use the symbol⊕to denote the addition operation for this
system in order to avoid confusion with the usual addition x+y of
row vectors.
Show that S, together with the ordinary scalar multiplication
and the addition operation⊕, is not a vector space.
Test ALL of the eight axioms and report which axioms
fail to hold.
Let S be the set of all integers x > 6543 such that the decimal representation...
Let S be the set of all integers x > 6543 such that the
decimal representation of x has distinct digits, none of which is
equal to 7, 8, or 9. (The decimal representation does not have
leading zeros.) Determine the size of the set S.
(do not just write out all elements of S.)
Let L be the set of all languages over alphabet {0}. Show that L is uncountable,...
Let L be the set of all languages over alphabet {0}. Show that L
is uncountable, using a proof by diagonalization.
Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a...
Let X be a non-degenerate ordered set with the order topology. A
non-degenerate set is a set with more than one
element.
Show the following:
(1) every open interval is open, (2) every closed interval is
closed, (3) every open
ray is open, and (4) every closed ray is closed.
Please note: Its a topology question.
Let S be the set of all integers x ∈ {1,2,...,100} such that the decimal representation...
Let S be the set of all integers x ∈ {1,2,...,100} such that the
decimal representation of x does not contain the digit 4. (The
decimal representation does not have leading zeros.) • Determine
the size of the set S without using the Complement Rule. • Use the
Complement Rule to determine the size of the set S. (You do not get
marks if you write out all numbers from 1 to 100 and mark those
that belong to the...
1)Let S be the set of all students at a college. Define a relation on the...
1)Let S be the set of all students at a college. Define a
relation on the set S by the rule that two people are related if
they live less than 2 miles apart. Is this relation an equivalence
relation on S? Justify your answer.
2) Define another relation on the set S from problem 5 by
defining two people as related if they have the same classification
(freshman, sophomore, junior, senior or graduate student). Is this
an equivalence relation...
1)Let S be the set of all students at a college. Define a relation on the...
1)Let S be the set of all students at a college. Define a
relation on the set S by the rule that two people are related if
they live less than 2 miles apart. Is this relation an equivalence
relation on S? Justify your answer.
2) Define another relation on the set S from problem 5 by
defining two people as related if they have the same classification
(freshman, sophomore, junior, senior or graduate student). Is this
an equivalence relation...
Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped...
Let X be the space of all continuous functions from [0, 1] to
[0, 1] equipped with the sup metric. Let Xi be the set of injective
and Xs be the set of surjective elements of A and let Xis = Xi ∩
Xs. Prove or disprove: i) Xi is closed, ii) Xs is closed, iii) Xis
is closed, iv) X is connected, v) X is compact.
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