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PHI 323 / MAT 313
PROBLEM SETS
Spring 2008
Problem Set #1 (Due WED 13 FEB)
Axioms of Set Theory. Algebra of Sets
1.
a.
b.
Show that if x  {x} = y  {y} then x = y.
Show that if a ≠ b then if {{x, a}, {y, b}} = {{u, a}, {v, b}}
then x = u and y = v.
2.
If a and b are subsets of I how many different sets at most can be obtained from
them using the operations  and  and - (where -x means I-x)?
Under what condition is x Y = {x  y: y  Y}?
Under what condition is x Y = {x  y: y  Y}?
3.
a.
b.
4.
Suppose r is a binary relation.
a.
Prove r[a  b] = r[a]  r[b]
b.
Prove r[a  b]  r[a]  r[b]
c.
d.
e.
5.
Prove r[a - b]  r[a] - r[b]
Give an example to show that we do not in general have equality in (b).
Give an example to show that we do not in general have equality in (c).
Let R be a set of binary relations.
a.
What is the relationship between dom R and {dom r: r  R}?
b.
What is the relationship between dom R and {dom r: r  R}?
2
Problem Set #2 (Due WED 20 FEB)
Relations (functions, equivalences, orders).
1.
Prove the generalized distributive law
i I Ai  j J Bj =  i, j  IJ (Ai  Bj )
2.
Given an equivalence relation E, we get from it a partition PE whose pieces are
the E-equivalence classes. Inversely, given an partition P we get from an equivalence
relation EP namely the relation of belonging to the same piece of the partition.
a.
Show that PEP = P.
b.
Show that EPE = E.
3.
Suppose that <A and <B are linear orderings on sets A and B respectively, let C =
A  B, and let <C be the following relation on C: a1, b1 <C a2, b2 iff either a1 <A a2
or a1 = a2 and b1 <B b2. Show that <C is a linear ordering on C.
4.
Prove that there is no function f from natural numbers to natural numbers such
that f(n) > f(n + 1) for all n.
5.
Let R be a binary relation on natural numbers. Suppose that for all m and n, if Rij
holds for all i and j with i < m or with i = m and j < n, then Rmn holds. Show that Rmn
holds for all m and n.
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Problem Set #3 (Due WED 27 FEB)
Natural numbers.
1.
a.
Prove the associative law for addition (x + y) + z = x + (y + z) using the
defining equations for + and induction.
b.
Prove the commutative law for addition x + y = y + x sing the defining
equations for + and induction.
The usual laws for multiplication (associative, commutative, distributive) can be proved
the same way.
2.
a.
Let m and n be natural numbers with m < n. Prove that there is a natural
number p ≠ 0 such that m + p = m.
b.
Prove that for any natural numbers x, y with y ≠ 0, there exist natural
numbers q, r with r < y such that x = y  q + r.
You may use the usual laws for addition and multiplication (those mentioned in the
preceding problem).
3.
Assume A is finite and let f be a function from A to A. Show that f is one-to-one iff
range f = A.
4.
a.
Prove that the Cartesian product of two finite sets A and B is finite, and
that |AB| = |A||B|.
b.
Prove that the power set of a finite set A is finite, and that |(A)| = 2|A|.
5.
Let A be a finite set and let < and  each linearly order A, Show that the two linear
orders are isomorphic.
4
Problem Set #4 (Due WED 05 MAR)
Real numbers.
1.
A sequence an: n  N of natural numbers is eventually constant iff for some m,
an = am for all n > m. Show that the set of eventually constant sequences of natural
numbers is countable.
2.
Let an, bn,: n  N be a sequence of pairs of natural numbers. Show that there
exist m < n such that am ≤ an and bm ≤ bn.
3.
Show that there is an uncountable set A of infinite sets of natural numbers such
that the intersection of any two sets in A is finite. (Hint: first prove this not for the natural
numbers but for the set of finite sequences of natural numbers.)
4.
For the integers, prove the distributive law for subtraction, x  (y - z) = x  y - x  z
using the laws of the arithmetic of natural numbers.
5.
Construct a one-to-one correspondence between the open interval
(0,1) = {x: x is a real number and 0 < x < 1}
and the closed interval
[0,1] = {x: x is a real number and 0 ≤ x ≤ 1}
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Problem Set #5 (Due WED 16 MAR)
Ordinal numbers.
1.
Show that if < and its inverse > are both well orderings of a set A, then A is finite.
2.
Prove the associative law for ordinal multiplication.
a.
Prove the distributive law  ( + ) =   +  
b.
Show that the distributive law ( + ) =  + 
does not hold in general.
3.
4.
a.
b.
5.
a.
Prove the law of exponents + =   
Prove the law of exponents () = 
Show that the cancellation law for addition
if  +  =  +  then  = 
does not hold in general.
b.
Show that the cancellation law for multiplication
if  > 0 and   =   then  = 
does not hold in general.
6
Problem Set #6 (due WED 02 APR)
Axiom of choice.
1.
Let A be a nonempty set and R a binary relation on A, and assume that for every x
in A there is a y in A such that xRy. Such that there is a function f from the natural
numbers to A such that f(n)Rf(n+1) for all n. This fact is known as the principle of
dependent choice.
2.
Let A be a nonempty set such that for every b, b is in A iff every finite subset of b
is in A. (This is called having finite character.) Show that A as an element b that is not a
subset of any other element of A. (This is called a maximal element.) This fact is known
as the Teichmüller-Tukey principle.
3.
Let *(A) be the set of all finite subsets of A. Show that if A is infinite, then
|A| = |*(A)|.
4.
Let  and  be transfinite cardinals. Assume that 2 ≤  ≤ 2. Show that  = 2.
5.
A permutation of a set A is a one-to-one function from A onto A. Let A! be the set
of all permutations of A. Show that if A is infinite, then |A!| = 2|A|.
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Problem Set #7 (due WED 09 APR)
Miscellany.
1.
a.
A G set is one that is an intersection of countably many open sets. How
many G sets of real numbers are there?
b.
An F set is one that is a union of countably many closed sets. How many
F sets of real numbers are there?
2.
Let  be a transfinite cardinal. If C are subsets of  for  < , then their
diagonal intersection is the set C = { < :   C for all  < }. Show that if the C are
closed unbounded, so is C.
3.
a.
Call a function f from countable ordinals to countable ordinals monotone
continous if f() < f() whenever  < , and if f() = sup{ f(). < } if  is a limit
ordinal. Call  a fixed point of f if f() = . Show that the set of fixed points of a
monotone continuous function is a closed unbounded set.
b.
If A is an unbounded set of countable ordinals, the enumerator of A is the
function f from countable ordinals defined by letting f() = the least element of A greater
than f() for all < . Show that if A is closed unbounded, its enumerator is monotone
continuous.
4.
a.
Show that if A has at least 9 elements and the set [A]2 of (unordered) pairs
of elements of A is partitioned into two pieces P and Q, then either there is a subset B of
A with at least 4 elements such that all pairs in [B]2 belong to P or else there is a subset C
of A with at least 3 elements such that all pairs in [C]2 belong to Q.
b.
Show that if A has at least 18 elements and [A]2 is partitioned into two
pieces, then there exists a subset B of A with at least 5 elements such that all pairs in [B]2
are in the same piece.
5.
A finite set of points in the plane, no point in the set lying on the line connecting
any two other points in the set, is said to be convex if no point lies in the inside of any
triangle whose vertices are also points in the set. It is a known result of geometry that any
set of at least 5 points in the plane, no point in the set lying on the line connecting any
two other points in the set, there is a subset of at least 4 points that is convex. Show that
for any m there is an n such that for any set of at least n points in the plane, no point in the
set lying on the line connecting any two other points in the set, there is a subset of at least
m points that is convex. This fact is known as the Erdös-Szekeres theorem. (Hint: This is
not a geometry problem; you have been given the only fact from geometry that you need.)
8
Problem Set #8 (due WED 16 APR)
Axiomatics.
1.
Let I be a nonempty set, A a nonempty set of subsets of I. Then A is called a field
of sets iff whenever one of the operations  or  or - (where -a means I-x) is applied to an
element or elements of A, the result is an element of A. For any nonempty set B of subsets
of I, A is called the field of sets generated by B if A is a field of sets, B  A and for any
other A' such that A' is a field of sets and B  A', we have A  A'. Prove that for any B the
field of sets generated by B exists, using the power axiom but not the replacement axiom.
2.
Prove the same result using the replacement axiom but not the power axiom.
3.
A binary relation R on a nonempty set A is well-founded if every nonempty subset
B of A has at least one R-minimal element, meaning an element a  B such that there is
no b  B with bRa and b ≠ a. An infinite decreasing sequence for R is a sequence an: n
 N of elements of a such that an+1R an for all n. Show that R is well-founded iff there is
no infinite decreasing sequence for R.
4.
A ranking for R is a function whose domain is A and values are ordinals, such that
whenever aRb, f(a) < f(b). Show that R is well-founded iff there is a ranking of R.
5.
Show that the following are expressible by 0=0=0 formulas:
a.
a is a set, r is a binary relation on a
b.
[a is a set, r is a binary relation on a and] xry
c.
a is a set, r is a binary relation on a, and r is irreflexive
d.
a is a set, r is a binary relation on a, and r is a linear order
e.
f is a function from a to b
f.
a and b are sets, r and s are linear orderings on a and b (respectively)
and f is an isomorphism from the former to the latter
9
Problem Set #9 (due WED 23 APR)
Axiomatics.
1.
Let  be any formula of the language of set theory. Use the replacement axiom to
prove the following collection principle:
uv(xu)(y (x, y) y v)(x, y))
2.
Let  be any formula of the language of set theory. Use the foundation axiom
following principle of -induction:
x((yx)(y) (x)) x (x)
3.
Show that if we dropped the foundation axiom and assumed -induction instead,
we could prove the foundation axiom.
4.
Show that the following notions are expressible by a 1 formulas:
a.
b.
5.
a is a nonempty set, r is a well-founded relation on a
a is a nonempty set, r is a non-well-founded relation on a
Let M be a transitive set that is model of ZF. Let bM. Show:
a.
M |= (b is a well-order) iff b is indeed a well-order
b.
M |= (b is a finite) iff b is indeed finite