Question

(a) Let A ⊂ R be open and B ⊂ R. Define AB = {xy ∈ R : x ∈ A and y ∈ B}. Is AB necessarily open? Why?

(b) Let S = {x ∈ R : x is irrational}. Is S closed? Why?

Thank you!

Answer #1

1. Let R be the rectangle in the xy-plane bounded by the lines x
= 1, x = 4, y = −1, and y = 2. Evaluate Z Z R sin(πx + πy) dA.
2. Let T be the triangle with vertices (0, 0), (0, 2), and (1,
0). Evaluate the integral Z Z T xy^2 dA
ZZ means double integral. All x's are variables. Thank you!.

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

let g be a group. let h be a subgroup of g. define
a~b. if ab^-1 is in h. prove ~ is an equivalence relation on g

let A = {−4, 4, 5, 8} and B = {4, 5, 6} and define relations R
and S from A to B as follows:
For all elements (x in A , y in B) , x R y ⇔ |x| = |y| + 1 and x
S y ⇔ x /y is an integer.
1. Find A X B and A intersect B.
2. Is the relation R reflexive ? Justify your answer.

Fix a positive real number c, and let S = (−c, c) ⊆ R. Consider
the formula x ∗ y :=(x + y)/(1 + xy/c^2).
(a)Show that this formula gives a well-defined binary operation
on S (I think it is equivalent to say that show the domain of x*y
is in (-c,c), but i dont know how to prove that)
(b)this operation makes (S, ∗) into an abelian group. (I have
already solved this, you can just ignore)
(c)Explain why...

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

Let A,B ∈ M3(R) with |A| = 4 and |B| = −3.
Evaluate (a) |AB|, (b) |5A|, (c) |BT|, (d) |A−1|, (e) |A3|, (f)
|ATA|, (g) |B−1AB|.

Let V=Mn(R) and <A,B>=tr(AB) be a symmetric bilinear form
on V. Determine the signature of tr(AB) for arbitrary n.

Let matrices A,B∈Mn×n(R). Show that if A and B are each similar
to some diagonal matrix, and also have the same eigenvectors (but
not necessarily the same eigenvalues), then AB=BA.

Let f:[0,1]——>R be define by f(x)= x if x belong to rational
number and 0 if x belong to irrational number and let g(x)=x
(a) prove that for all partitions P of [0,1],we have
U(f,P)=U(g,P).what does mean about U(f) and U(g)?
(b)prove that U(g) greater than or equal 0.25
(c) prove that L(f)=0
(d) what does this tell us about the integrability of f ?

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