Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if and
only if ad=bc. (a) Show that R is an equivalence relation. (b) What
is the equivalence class of (1,2)? List out at least five elements
of the equivalence class. (c) Give an interpretation of the
equivalence classes for R. [Here, an interpretation is a
description of the equivalence classes that is more meaningful than
a mere repetition of the definition of R. Hint:...
9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R.
(i) List explicitly all the ideals A of R with I ⊆ A.
(ii) Write out all the elements of R/I (these are cosets).
(iii) List explicitly the set of all ideals B of R/I (these are
sets of cosets).
(iv) Let π: R → R/I be the natural projection. For each ideal A
of R such that I ⊆ A, write out π(A) explicitly...
Let G = Z x Z and H = {(a, b) in Z x Z | 8 divides a+b}
a. Prove directly that H is a normal subgroup in G (use the fact
that closed under composition and inverses)
b. Prove that G/H is isomorphic to Z8.
c. What is the index of [G : H]?
Integral
Let f:[a,b]→R and g:[a,b]→R be two bounded functions.
Suppose f≤g on [a,b]. Use the information to prove
thatL(f)≤L(g)andU(f)≤U(g).
Information:
g : [0, 1] —> R be defined by if
x=0, g(x)=1; if x=m/n (m and n are positive
integer with no common factor), g(x)=1/n; if x
doesn't belong to rational number, g(x)=0
g is discontinuous at every rational number
in[0,1].
g is Riemann integrable on [0,1] based on the fact that
Suppose h:[a,b]→R is continuous everywhere except at a...
6. Let R be a relation on Z x Z such that for all ordered pairs
(a, b),(c, d) ∈ Z x Z, (a, b) R (c, d) ⇔ a ≤ c and b|d . Prove that
R is a partial order relation.
(V) Let A ⊆ R, B ⊆ R, A 6= ∅, B 6= ∅ be two bounded subset of R.
Define a set A − B := {a − b : a ∈ A and b ∈ B}. Show that sup(A −
B) = sup A − inf B and inf(A − B) = inf A − sup B