HW21
1. Prove that any set of triangles in R2 whose vertices have rational coordinates is countable.
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Solution. Let T be a triangle with vertices (r1 ; r1 ) ; (r2 ; r2 ) and
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(r3 ; r3 ), where rj ; rj 2 Q, j = 1; 2; 3. We can always assume that
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Real Analysis - Homework solutions
Chris Monico, May 2, 2013
1.1 (a) Rings (resp. -rings) are closed under finite (resp. countable) intersections.
(b) If R is a ring (resp. -ring) then R is an algebra (resp. -algebra) iff X R.
(c) If R is a (nonempty) -ri
1. (a) Prove that, if a subset E in metric space (M; ) is of the rst category and A E , then A is also of the rst category.
(b) Prove that if (En )1
(M; ) is a sequence of sets of the rst
n=1
1
category, then E = [n=1 En is also of the rst category.
2. (a
Real analysis for graduate
students
Version 3.1
Richard F. Bass
January 6, 2016
ii
c Copyright 2016 Richard F. Bass
All rights reserved.
First edition published 2011.
Second edition published 2013.
Version 3.1 published 2016.
iii
To the memory of my paren
onto
1. (a) Let (A; ) be a well-ordered set, B
A and f : A ! B be an
order isomorphism of (A; ) onto (B; ). Prove that for every a 2 A,
a f (a).
(b) Let (A; ) be a well-ordered set and a 2 A. Recall that the set
Ia = fx 2 A j x
ag
is called an initial seg
HW 31
onto
1. (a) Let (A; ) be a well-ordered set, B
A and f : A ! B be an
order isomorphism of (A; ) onto (B; ). Prove that for every a 2 A,
a f (a).
(b) Let (A; ) be a well-ordered set and a 2 A. Recall that the set
Ia = fx 2 A j x
ag
is called an initi
HW4
1. Let X be an uncountable set and R X be the family of all sets which
either countable or their complements (in X) are countable. Prove that
R is a ring.
2. Describe the minimal ring generated by F in the following cases:
Describe the minimal ring ge
HW2
1. Prove that any set of triangles in R2 whose vertices have rational coordinates is countable.
Note: Let x1 ; x2 ; x3 2 R2 . Then the triangle T with vertices x1 ; x2 ; x3
is the union of points of the segments [x1 ; x2 ] ; [x2 ; x3 ] and [x1 ; x3 ].