Question

1. a) Let f : C → D be a function. Prove that if C_{1}
and C_{2} be two subsets of C, then

f(C_{1}ꓴC_{2}) = f(C_{1}) ꓴ
f(C_{2}).

b) Let f : C → D be a function. Let C_{1} and C_{2}
be subsets of C. Give an example of

sets C, C_{1}, C_{2} and D for which f(C ꓵ D) ≠
f(C_{1}) ꓵ f(C_{2}).

Answer #1

Let C1 be a circle with radius a C2 a circle with radius b, and
let the centers of C1 and C2 have a distance of c apart.
In terms of a,b, and c prove:
When C1 n C2 when are there zero intersection points? Prove
this
When are there two intersection points? Prove this
When are three intersection points? Prove this

let f:A->B and let D1, D2, and D be subsets of A.
Prove or Disprove
F^-1(D1UD2)=F^-1(D1)UF^-1(D2)

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

Let f : A → B, and let V ⊆ B.
(a) Prove that V ⊇ f(f−1(V )).
(b) Give an explicit example where the two sides are not
equal.
(c) Prove that if f is onto then the two sides must be
equal.

Let E and F be two disjoint closed subsets in metric space
(X,d). Prove that there exist two disjoint open subsets U and V in
(X,d) such that U⊃E and V⊃F

Let f : A → B be a function and let A1 and A2 be subsets of A.
Prove that if f is one-to-one, then f(A1 ∩ A2) = f(A1) ∩ f(A2).

Let Z be the integers.
(a) Let C1 = {(a, a) | a ∈ Z}. Prove that
C1 is a subgroup of Z × Z.
(b) Let n ≥ 2 be an integer, and let Cn = {(a, b) | a
≡ b( mod n)}. Prove that Cn is a subgroup of Z × Z.
(c) Prove that every proper subgroup of Z × Z that contains
C1 has the form Cn for some positive integer
n.

Let A, B, C and D be sets. Prove that A\B ⊆ C \D if and only if
A ⊆ B ∪C and A∩D ⊆ B

Let A, B, C and D be sets. Prove that A \ B and C \ D are
disjoint if and only if A ∩ C ⊆ B ∪ D.

Let S be the set of all real circles, defined by (x-a)^2 +
(y-b)^2=r^2. Define d(C1,C2) =
√((a1 − a2)^2 + (b1 −
b2)^2 + (r1 − r2)^2 so that S is a
metric space.
Prove that metric space S is NOT a complete metric space. Give a
clear example.
Describe points of C\S as limits of the appropriate sequences of
circles, where C is the completion of S.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 6 minutes ago

asked 8 minutes ago

asked 12 minutes ago

asked 18 minutes ago

asked 18 minutes ago

asked 18 minutes ago

asked 30 minutes ago

asked 38 minutes ago

asked 50 minutes ago

asked 54 minutes ago

asked 57 minutes ago