# Introductory real analysis: exercise 3:

Based on previous two blogs. (Reference: Introductory Real Analysis by Kolmogorov and Fomin, Dover Publishers):

Problem 1:Exhibit both a partial ordering and a simple ordering of the set of all complex numbers.

Problem 2:What is the minimal element of the set of all subsets of a given set X, partially ordered by set inclusion. What is the maximal element?

Problem 3: A partially ordered set M is said to be a directed set if, given any two elements $a, b \in M$, there is an element $c \in M$ such that $a < c, b. Are the partially ordered sets in the previous blog(s) Section 3.1 all directed sets?

Problem 4: By the greatest lower bound of two elements a and b of a partially ordered set M, we mean an element $c \in M$ such that $c \leq a, c \leq b$ and there is no element $d \in M$ such that $a < d \leq a, d \leq b$. Similarly, by the least upper bound of a and b, we mean an element $c \in M$ such that $a \leq c, b \leq c$ and there is no element $d \in M$ such that $a \leq d . By a lattice is meant a partially ordered set any two elements of which have both a greatest lower bound and a least upper bound. Prove that the set of all subsets of a given set X, partially ordered by set inclusion, is a lattice. What is the set theoretic meaning of the greatest lower bound and least upper bound of two elements of this set?

Problem 5: Prove that an order preserving mapping of one ordered set onto another is automatically an isomorphism.

Problem 6: Prove that ordered sums and products of ordered sets are associative, that is, prove that if $M_{1}, M_{2}, M_{3}$ are ordered sets, then

$(M_{1}+M_{2})+M_{3}=M_{1}+(M_{2}+M_{3})$,

$(M_{1}.M_{2}).M_{3}=M_{1}.(M_{2}.M_{3})$ where the operations + and . are the same as defined in previous blog(s).

Comment: This allows us to drop the parentheses in writing ordered sums and products.

Problem 7:

Construct well-ordered sets with ordinals

$\omega + n$, $\omega + \omega$, $\omega + \omega + n$, $\omega + \omega + \omega$, $\ldots$.

Show that the sets are all countable.

Problem 8:

Construct well-ordered sets with ordinals

$\omega . n$, $\omega^{2}$, $\omega^{2}.n$, $\omega^{3}$, $\ldots$.

Show that the sets are all countable.

Problem 9:

Show that $\omega + \omega = \omega. 2$, $\omega + \omega + \omega = \omega. 3$, $\ldots$

Problem 10:

Prove that the set $W(\alpha)$ of all ordinals less than a given ordinal $\alpha$ is well-ordered.

Problem 11:

Prove that any non-empty set of ordinals is well-ordered.

Problem 12:

Prove that the set M of all ordinals corresponding to a countable set is itself uncountable.

Problem 13:

Let $\aleph_{1}$ be the power of the set M in the previous problem. Prove that there is no power m such that $\aleph_{0} .

More later,

Nalin Pithwa.

# Introductory Real Analysis: The well-ordering theorem, the axiom of choice,equivalent assertions

Reference: Introductory Real Analysis by Kolmogorov and Fomin, Dover Publications.

Section 3.7 The well-ordering theorem, the axiom of choice, and equivalent assertions:

Theorem 4 of previous blog shows that the powers of two well-ordered sets are always comparable. In 1904, Zermelo succeeded in proving the

Well-ordering theorem: Every set can be well-ordered.

It follows from the well-ordering theorem and Theorem 5 (of previous blog) that the powers of two arbitrary sets are always comparable, a fact already used earlier. Zermelo’s proof, which will not be given here, rests on the following basic:

AXIOM OF CHOICE:

Given any set M, there is a “choice function” f such that $f(A)$ is an element of A for every non-empty subset $A \subset M$.

We will assume the validity of the axiom of choice without further ado. In fact, without the axiom of choice we would be severely hampered in making set-theoretic constructions. However, it should be noted that from the standpoint of the foundations of set theory, there are still deep and controversial problems associated with the use of axiom of choice.

There are a number of assertions equivalent to the axiom of choice, that is, assertions each of which both implies and is implied by the axiom of choice. One of these is the well-ordering theorem, which obviously implies the axiom of choice. In fact, if an arbitrary set M can be well-ordered, then, by merely choosing the “first” element in each subset $A \subset M$, we get the function $f(A)$ figuring in the statement of the axiom of choice. On the other hand, the axiom of choice implies the well-ordering theorem, as already noted without proof.

To state further assertions equivalent to the axiom of choice, we need some more terminology:

DEFINITION 3:

Let M be a partially ordered set, and let A be any subset of M such that a and b are comparable for every $a, b \in A$. Then, A is called a chain (in M). A chain C is said to be maximal if there is no other chain C in M containing C on a proper subset.

DEFINITION 4:

An element a of a partially ordered set M is called an upper bound of a subset $M^{'} \subset M$, if $a^{'} < a$ for every $a^{'} \in M^{'}$.

We now have the vocabulary needed to state two other assertions equivalent to the axiom of choice:

Hausdorff’s Maximal Principle: Every chain in a partially ordered set M is contained in a maximal chain in M.

Zorn’s lemma:  If every chain in a partially ordered set M has an upper bound, then M contains a maximal element.

For the proof of the equivalence of the axiom of choice, the well-ordering theorem, Hausdorff’s maximal principle and Zorn’s lemma, we refer the reader elsewhere. Of these various equivalent assertions, Zorn’s lemma is perhaps the most useful.

Section 3.8 Transfinite Induction:

Mathematical propositions are very often proved by using the following familiar:

THEOREM 4: MATHEMATICAL INDUCTION:

Given a proposition $P(n)$ formulated for every positive integer n, suppose that :

i) P(1) is true.

ii) The validity of P(k) for all $k implies the validity of $P(n+1)$. Then, P(n) is true for all $n=1,2,\ldots$.

PROOF 4:

Suppose P(n) fails to be true for all $n=1,2,\ldots$ and let n, be the smallest integer for which $P(n)$ is false (the existence of such $n_{1}$ follows from the well-ordering of the positive integers). Clearly, $n_{1}>1$, so that $n_{1}-1$ is a positive integer. Therefore, $P(n)$ is valid for all $k \leq n_{1}-1$ but not for $n_{1}$. Contradiction!!

QED.

Replacing the set of all positive integers by an arbitrary well-ordered set, we get

THEOREM 4 (Transfinite induction):

Given a well-ordered set A, let $P(a)$ be a proposition formulated for every element $a \in A$. Suppose that

i) $P(a)$ is true for the smallest element of A

ii) The validity of $P(a)$ for all $a < a^{*}$ implies the validity of $P(a^{*})$. Then, $P(a)$ is true for all $a \in A$.

PROOF 4:

Suppose $P(a)$ fails to be true for all $a \in A$. Then, $P(a)$ is false for all a in some non empty subset $A^{*} \subset A$. By the well-ordering, $A^{*}$ has a smallest element $a^{*}$. Therefore, $P(a)$ is valid for all $a< a^{*}$ but not for $a^{*}$. Contradiction !!

QED.

Remark:

Since any set can be well-ordered, by the well-ordering theorem, transfinite induction can in principle be applied to any set M whatsoever. In practice, however, Zorn’s lemma is a more useful tool, requiring only that M be partially ordered.

Section 3.9: Historical Remarks:

Set theory as a branch of mathematics in its own right stems from the pioneer work of Georg Cantor (1845-1918). Originally met with disbelief, Cantor’s ideas subsequently became widespread. By now, the set theoretic point of view has become standard in the most diverse fields of mathematics. Basic concepts, like groups, rings, fields, linear spaces etc are habitually defined as sets of elements of an arbitrary kind obeying appropriate axioms.

Further development of set theory led to a number of logical difficulties, which naturally gave rise to attempts to replace “naive” set theory by a more rigorous axiomatic set theory. It turns out that certain set theoretic questions, which would at first seem to have “yes” or “no” answers, are in fact of a different kind. Thus, it was shown by Godel in 1940 that a negative answer to the question :”Is there an uncountable set of power less than that of the continuum” is consistent with set theory (axiomatized in a way we will not discuss here), but it was recently shown by Stephen Cohen that an affirmative answer to the question is also consistent in the same sense !!

We will continue to exercises based on last two blogs now,

Regards,

Nalin Pithwa.

# Ordered Sets and Ordinal Numbers: Introductory Real Analysis: Kolmogorov and Fomin

Section 3: Ordered sets and ordinal numbers:

Section 3.1: Partially ordered sets.

A binary relation R on a set M is said to be a partial ordering (and the set M is said to be partially ordered) if

i) R is reflexive (aRa for every $a \in M$).

ii) R is transitive (aRb and bRc together imply aRc)

iii) R is antisymmetric in the sense that aRb and bRa together imply $a=b$.

For example, if M is the set of all real numbers and aRb means that $a, then R is partial ordering. (please verify!) This suggests writing $a \leq b$ (or equivalently $b \geq a$) instead of aRb whenever R is a partial ordering, and we will do so from now on. Similarly, we write $a < b$ if $a \leq b$, but $a \neq b$; and $b > a$ if $b \geq a$ but $b \neq a$.

The following examples give some idea of the generality of the concept of a partial ordering:

Example 1:

Any set M can be trivially partially ordered by setting $a \leq b$ iff $a=b$.

Example 2:

Let M be the set of all continuous functions $f, g, \ldots$ defined in a closed interval $[\alpha, \beta]$. Then, we get a partial ordering by setting $f \leq g$ iff $f(t) \leq g(t)$ for every $t \in [\alpha, \beta]$.

Example 3:

The set of all subsets $M_{1}, M_{2}, \ldots$ is partially ordered if and only if $M_{1} \leq M_{2}$ means that $M_{1} \subset M_{2}$.

Example 4:

The set of all integers greater than 1 is partially ordered if “$a \leq b$” means that “b is divisible by a.”

Remark: An element of a of a partially ordered set is said to be maximal if $a \leq b$ implies $b=a$ and minimal if $b \leq a$ implies $b=a$. Thus, in Example 4 above, every prime number (greater than 1) is a minimal element. (Of course, 1 is neither prime nor composite.)

Section 3.2: Order-preserving mappings. Isomorphisms:

Let M and $M^{'}$ be any two partially ordered sets, and let f be a one-to-one mapping of M onto $M^{'}$. Then, f is said to be order-preserving if $a \leq b$ (where $a, b \in M$) implies $f(a) \leq f(b)$ (in $M^{'}$). An order preserving mapping f such that $f(a) \leq f(b)$ implies $a \leq b$ is called an isomorphism. In other words, an isomorphism between two partially ordered sets M and $M^{'}$ is a one-to-one mapping of M onto $M^{'}$ such that $f(a) \leq f(b)$ if and only if $a \leq b$. Two partially ordered sets M and $M^{'}$ are said to be isomorphic (to each other) if there exists and isomorphism between them.

Example:

Let M be the set of positive integers greater than 1 partially ordered as in example 4 earlier. Let $M^{'}$ be the same set partially ordered in the natural way, that is, in such a way that $a \leq b$ if and only if $b-a >0$. Then, the mapping of M onto $M^{'}$ carrying every integer n into itself is order-preserving, but NOT an isomorphism.

Some remarks:

Isomorphism between partially ordered sets is an equivalence relation as defined in an earlier blog of this series, being obviously reflexive, symmetric, and transitive. Hence, any given family of partially ordered sets can be partitioned into disjoint classes of isomorphic sets. Clearly, two isomorphic partially ordered sets can be regarded as identical in cases where it is the structure of the partial ordering than the specific nature of the elements of the sets that is of interest.

Some other remarks:

For those of you who are aware, compare now the concepts and some examples of homomorphisms and isomorphisms between two algebraic objects called groups with the above concepts.  Two references for the same are: (i) Topics in Algebra by I N Herstein (2) Abstract Algebra by Dummit and Foote.

Some further remarks:

At this stage, it might even be useful to compare the concepts and examples of a cartesian product of two (non-empty) sets A and B; that of a relation R on set A and set B; that of a binary relation on a set A; that, of an equivalence relation on a set A; and also the basic meaning of a function from set A to set B. You will see that there are slight subtle differences.

Sec 3.3 Ordered Sets. Order Types:

Given two elements a and b of a partially ordered set M, it may turn out that neither of the relations $a \leq b$ or $b \leq a$ holds. In this case, a and b are said to be noncomparable. Thus, in general, the relation $\leq$ is defined only for certain pairs of elements, which is why M is said to be partially ordered. However, if M has no non-comparable elements, then M is said to be simply ordered or linearly ordered or ordered. In other words, a set M is ordered if it is partially ordered and if, given any two distinct elements $a, b \in M$, either $a , or $b. Obviously, any subset of an ordered set is itself ordered.

Each of the sets figuring in Examples 1 to 4 of Section 3.1 in this blog, is partially ordered, but not ordered. Simple examples of ordered sets are the set of all positive integers, the set of all rational numbers, the set of all real numbers in the $[0,1]$ and so on (with the usual relations of “greater than” and “less than”.)

Since an ordered set is a special kind of partially ordered set, the concepts of order-preserving mapping and isomorphism apply equally well to ordered sets. Two isomorphic ordered sets are said to have the same order type. Thus “type” is something shared by all isomorphic ordered sets, just as “power” is something shared by all equivalent sets (considered as ‘plain’ sets, without regard for possible orderings).

The simplest example of an ordered set is the set of all positive integers $1,2,3,\ldots$ arranged in increasing order, with the usual meaning of the symbol <. The order type of this set is denoted by the symbol $\omega$. Two isomorphic ordered sets obviously have the same power (because an isomorphism is a one-to-one correspondence). Thus, it makes sense to talk about the power corresponding to a given order type. The converse is not true, since a set of a given power can in general be ordered in many different ways. It is only in the finite case that the number of elements in a set uniquely determines its type, designated by the same symbol n as the number of elements in the set. For example, besides the “natural” order type $\omega$ of the set of positive integers, there is another order type corresponding to the sequence: $1,3,5, \ldots, 2,4,6, \ldots$ where odd and even numbers are separately arranged in increasing order but any odd number precedes any even number. It can be shown that the number of distinct order types of a set of power $\aleph_{0}$ is infinite and in fact, uncountable.

Section 3.4 Ordered sets and products of ordered sets.

Let $M_{1}$ and $M_{2}$ be two ordered sets of types $\theta_{1}$ and $\theta_{2}$ respectively. Then, we can introduce an ordering in the union $M_{1} \bigcup M_{2}$ of the two sets by assuming that :

(i) a and b have the same ordering as in $M_{1}$ if $a,b \in M_{1}$.

(ii) a and b have the same ordering as in $M_{2}$ if $a,b \in M_{2}$

(iii) $a if $a \in M_{1}$ and $b \in M_{2}$.

(Verify that this is actually an ordering of $M_{1} \bigcup M_{2}$). The set $M_{1} \bigcup M_{2}$ ordered in this way is called the ordered sum of $M_{1}$ and $M_{2}$ denoted by $M_{1}+M_{2}$. Note that the order of terms matters here, that is, in general $M_{2}+M_{1}$ is not isomorphic to $M_{1}+M_{2}$. More generally, we can define the ordered sum of any finite number of ordered sets by writing (this will be an exercise problem after this blogged section) :

$M_{1}+M_{2}+M_{3}=(M_{1}+M_{2})+M_{3}$

$M_{1}+M_{2}+M_{3}+M_{4}=(M_{1}+M_{2}+M_{3})+M_{4}$,

and so on.

By the ordered sum of the types $\theta_{1}+\theta_{2}$, denoted by $\theta_{1}+\theta_{2}$ we mean the order type of the set $M_{1}+M_{2}$.

Example:

Consider the order types $\omega$ and $n$. It is easy to see that $n + \omega = \omega$. In fact, if finitely many terms are written to the left of the sequence $1,2,3,\ldots, k, \ldots$, we again get a set of the same type (Quiz: why?). On the other hand, the order type $\omega + n$, that is the order type of the set (***) :

$\{ 1,2,3, \ldots, k, \ldots, a_{1}, a_{2}, \ldots, a_{n}\}$ is obviously not equal to $\omega$.

(Note: ***: In the above listing of elements of the set, we have used the usual curly braces notation, but with the emphasis on the order as shown.)

Again, let $M_{1}$ and $M_{2}$ be two ordered sets of types $\theta_{1}$ and $\theta_{2}$ respectively, Suppose we replace each element of $M_{2}$ by a “replica” of the set $M_{1}$. Then, the resulting set, denoted  by $M_{1}.M_{2}$ is the set of all pairs $(a,b)$ where $a \in M_{1}$ and $a \in M_{2}$ ordered in such a way that:

(i) $(a_{1},b_{1})<(a_{2},b_{2})$ if $b_{1} for arbitrary $a_{1},a_{2}$

(ii) $(a_{1},b)<(a_{2},b)$ if $a_{1}.

Note that the order of factors matters here, that is, in general $M_{2}.M_{1}$ is not isomorphic to $M_{1}.M_{2}$. The ordered product of any finite number of ordered sets can be defined by writing (again a problem in the exercise later):

$M_{1}.M_{2}.M_{3}=(M_{1}.M_{2}).M_{3}$

$M_{1}.M_{2}.M_{3}.M_{4}=(M_{1}.M_{2}.M_{3}).M_{4}$

and so on. By the ordered product of the types $\theta_{1}$ and $\theta_{2}$, denoted by $\theta_{1}.\theta_{2}$, we mean the order type of the set $M_{1}.M_{2}$.

Section 3.5. Well-ordered sets. Ordinal numbers.

A key concept in the theory of ordered sets is given by:

DEFINITION 1: An ordered set M is said to be well-ordered if every non-empty subset A of M has a smallest (or, “first“) element, that is, an element $\mu$ such that $\mu  for every $a \in A$.

Example 1: Every finite ordered set is obviously well-ordered.

Example 2: Every non-empty subset of a well-ordered set is itself well-ordered.

Example 3: The set M of numbers in the interval $[0,1]$ is ordered but not well-ordered. It is true that M has a smallest element, namely, the number 0 but the subset of M consisting of all positive rational numbers has no smallest element.

DEFINITION 2: The order type of a well-ordered set is called an ordinal number or simply, an ordinal. If the set is infinite, the ordinal is said to be transfinite. (NB: This is a good place to point out that the terms “cardinal number” and “power” of a set are synonymous).

Example 4: The set of positive integers $1,2,\ldots, k, \ldots$ arranged in increasing order is well-ordered, and hence, its order type $\omega$ is a transfinite ordinal. The order type $\omega + n$ of the set $\{1,2,\ldots, k, \ldots, a_{1}, a_{2}, \ldots, a_{n} \}$ is also an ordinal.

Example 5: The set $\{ \ldots, -k, \ldots, -3,-2,-1\}$ is ordered but not well-ordered. It is true that any non empty subset A of this set has a largest element (that is, an element $\mathscr{v}$ such that $a ) for every $a \in A$), but in general set A will not have a smallest element. In fact, the given set itself has no smallest element. Hence, the order type of given set is denoted by $\omega^{*}$ is not an ordinal number.

THEOREM 1The ordered sum of a finite number of well-ordered sets $M_{1}, M_{2}, \ldots, M_{n}$ is itself a well-ordered set.

PROOF 1: Let M be an arbitrary subset of the ordered sum $M_{1}+M_{2}+\ldots+M_{n}$ and let $M_{k}$ be the first of the sets $M_{1}, M_{2}, \ldots, M_{n}$, (namely, the set with the smallest index) containing elements of M. Then, $M \bigcap M_{k}$ is a subset of the well-ordered set $M_{k}$ and as such has a smallest element $\mu$. Clearly, $\mu$ is the smallest element of M itself. QED.

COROLLARYThe ordered sum of a finite number of ordinal numbers is itself an ordinal number.

Thus, new ordinal numbers can be constructed from any given set of ordinal numbers. For example, starting from the positive integers, (that is, the finite ordinal numbers) and the ordinal number $\omega$, we can construct the new ordinal numbers :

$\omega + n$, $\omega + \omega$, $\omega + \omega + n$, $\omega + \omega + \omega$, and so on.

THEOREM 2The ordered product of two well-ordered sets $M_{1}, M_{2}$ is itself a well-ordered set.

Proof 2:

Let M be an arbitrary subset of $M_{1}\dot M_{2}$ so that M is a set of ordered pairs $(a,b)$ with $a \in M_{1}, b\in M_{2}$. The set of all second elements b of pairs in M is a subset of $M_{2}$, and as such has a smallest element since $M_{2}$ is well-ordered. Let $b_{1}$ denote this smallest element, and consider all pairs of the form $(a,b_{1})$ contained in M. The set of all first elements a of these pairs is a subset of $M_{1}$, and as such has a smallest element since $M_{1}$ is well-ordered. Let $a_{1}$ denote this smallest element. Then, the pair $(a,b)$ is clearly the smallest element of M. QED.

COROLLARY 1:

The ordered product of a finite number of well-ordered sets is itself a well-ordered set.

COROLLARY 2:

The ordered product of a finite number of ordinal numbers is itself an ordinal number.

Thus it makes sense to talk about ordinal numbers $\omega . n, \omega^{2}, \omega^{2} . n, \omega^{3}, \ldots$ and so on. It is also possible to define such ordinal numbers as $\omega^{\omega}, \omega^{\omega^{\omega}}, \ldots$.

Section 3.6 Comparison of ordinal numbers:

If $n_{1}$ and $n_{2}$ are two ordinal numbers, then they either coincide or else one is larger than the other. As we now show, the same is true for transfinite ordinal numbers. We begin by observing that every element $a$ of a well-ordered set M determines an (initial section) P, the set of all $x \in M$ such that $x, and a remainder Q, the set of all $x \in M$ such that $x \geq a$. Given any two ordinal numbers $\alpha$ and $\beta$, Let M and N be well-ordered sets of order type $\alpha$ and $\beta$. Then, we say that

i) $\alpha=\beta$ if M and N are isomorphic;

ii) $\alpha < \beta$ if M is isomorphic to some section of N

iii) $\alpha > \beta$ if N is isomorphic to some section of M.

(note that this definition makes sense for finite $\alpha$ and $\beta$).

LEMMA:

Let f be an isomorphism of a well-ordered set A onto some subset $B \subset A$. Then, $f(a) \geq a$ for all $a \in A$.

Proof of lemma:

If there are elements $a \in A$ such that $f(a), then there is a least such element since A is well-ordered. Let $a_{0}$ be this element, and let $b_{0}=f(a_{0})$. Then $b_{0}, and hence, $f(b_{0}), since f is an isomorphism. But, then $a_{0}$ is not the smallest element such that $f(a). Contradiction! QED.

It follows from the lemma that a well-ordered set A cannot be isomorphic to any of its sections, since if A were isomorphic to the section determined by a, then clearly $f(a). In other words, the two relations

$\alpha=\beta$, $\alpha<\beta$ are incompatible, and so are $\alpha=\beta$, $\alpha>\beta$. Moreover, the two relations $\alpha<\beta$, $\alpha>\beta$ are incompatible, since otherwise we could use the transitivity to deduce $\alpha < \alpha$, which is impossible the lemma. Therefore, if one of the three relations

$\alpha<\beta$, $\alpha=\beta$, $\alpha>\beta$….II

holds, the other two are automatically excluded. We must still show that one of the relations in II above always holds, thereby proving that any two ordinal numbers are always comparable.

THEOREM 3:

Two given ordinal numbers $\alpha$ and $\beta$ always satisfy one and only of the relations: $\alpha=\beta$, $\alpha < \beta$ and $\beta<\alpha$.

Proof 3:

Let $W(\alpha)$ be the set of all ordinals $<\alpha$. Any two numbers $\gamma$ and $\gamma^{'}$ in $W(\alpha)$ are comparable (Recall the meaning of $\gamma < \alpha, \gamma^{'}<\alpha$ and use the fact that a section of a section of a well-ordered set is itself a section of a well-ordered set) and the corresponding ordering of $W(\alpha)$ makes it a well-ordered set of type $\alpha$. In fact, if a set $A = \{ \ldots, a, \ldots, b, \ldots\}$ is of type $\alpha$, then by definition, the ordinals less than $\alpha$ are the types of well-ordered sets isomorphic to sections of A. Hence, the ordinals themselves are in one-one correspondence with the elements of A. In other words, the elements of a set of type $\alpha$ can be numbered by using the ordinals less than $\alpha$: $A = \{ a_{1}, a_{2}, \ldots, a_{n}, \ldots\}$.

Now, let $\alpha$ and $\beta$ be two ordinals. Then, $W(\alpha)$ and $W(\beta)$ are well-ordered sets of types $\alpha$ and $\beta$, respectively. Moreover, let $C=A \bigcap B$ be the intersection of the sets A and B, that is, the set of all ordinals less than both $\alpha$ and $\beta$. Then, C is well-ordered, of type $\gamma$, say. We now show that $\gamma \leq \alpha$. If $C=A$, then obviously $\gamma=\alpha$. On the other hand, if $C \neq A$, then C is a section of A and hence, $\gamma < \alpha$. In fact, let $\xi \in C$, $\eta \in A - C$. Then, $\xi$ and $\eta$ are comparable, that is, either $\xi < \eta$ or $\xi > \eta$. But, $\eta < \xi < \alpha$ is impossible, since then $\eta \in C$. Therefore, $\xi < \eta$ and hence, C is a section of A, which implies $\gamma < \alpha$. Moreover, $\gamma$ is the first element of the set $A-C$. Thus, $\gamma < alpha$, as asserted, and similarly, $\gamma < \beta$. The case $\gamma < \alpha$, $\gamma < \beta$ is impossible, since then $\gamma \in A-C$ and $\gamma \in B-C$. But then $\gamma \notin C$ on the one hand and $\gamma \in A \bigcap B = C$ on the other hand. It follows that there are only three possibilities:

$\gamma=\alpha$ and $\gamma = \beta$ and $\alpha=\beta$

$\gamma=\alpha$ and $\gamma< \beta$ and $\alpha < \beta$

$\gamma < \alpha$ and $\gamma = \beta$ and $\alpha > \beta$,

that is, $\alpha$ and $\beta$ are comparable.

QED.

THEOREM 4:

Let A and B be well-ordered sets. Then, either A is equivalent to B or one of the sets is of greater power than the other, that is, the powers of A and B are comparable.

PROOF 4:

There is a definite power corresponding to each ordinal. But we have just seen that ordinals are comparable, and so are the corresponding powers.

[[recall the definition of inequality of powers given in Sec 2.5 in the blog series of Kolmogorov and Fomin). (Reproduced here for the sake of helping the memory: Given any two sets M and N, suppose M and N are equivalent. Then, M and N are said to have the same power. Roughly speaking, “power” is something shared by equivalent sets. If M and N are finite, then M and N have the same number of elements, and the concept of the power of a set reduces to the usual notion of the number of elements in a set. The power of the set $\mathcal{Z_{+}}$ is $\aleph_{0}$ and sets which are equivalent to the set $[0,1]$ are said to have the power of the continuum, c. For the powers of finite sets, that is, for the set of positive integers, we have the notions of “greater than” or “less than”, as well as the notion of equality. These concepts can be extended to infinite sets. If A and B are two sets with powers m(A) and m(B) respectively, either $m(A)=m(B)$, or A is equivalent to a subset of B such that no subset of B is equivalent to A, then by analogy with the final case, m(A) is less than m(B).]]

To be continued soon,

Regards,

Nalin Pithwa

# Equivalence relations and partitions: some core basic theorems

Suppose R is an equivalence relation on a set S. For each a in S, let [a] denote the set of elements of S to which a is related under R, that is, $[a] = \{ x: (a,x) \in R\}$

We call [a] the equivalence class of a in S under R. The collection of all such equivalence classes is denoted by S/R, that is, $S/R = \{ [a]: a \in S\}$. It is called the quotient set of S by R.

The fundamental property of an equivalence relation and its quotient set is contained in the following theorem:

Theorem I:

Let R be an equivalence relation on a set S. Then, the quotient set S/R is a partition of S. Specifically,

(i) For each $a \in S$, we have $a \in [a]$.

(ii) $[a]=[b]$ if and only if $(a,b) \in R$.

(iii) If $[a] \neq [b]$, then [a] and [b] are disjoint.

Proof of (i):

Since R is reflexive, $(a,a) \in R$ for every $a \in S$ and therefore $a \in [a]$.

Proof of (ii):

Assume: $(a,b) \in R$.

we want to show that $[a] = [b]$. That is, we got to prove, (i) $[b] \subseteq [a]$ and (ii) $[a] \subseteq [b]$.

Let $x \in [b]$; then, $(b,x) \in R$. But, by hypothesis $(a,b) \in R$ and so, by transitivity, $(a,x) \in R$. Accordingly, $x \in [a]$. Thus, $[b] \subseteq [a]$.

To prove that $[a] \subseteq [b]$, we observe that $(a,b) \in R$ implies, by symmetry, that $(b,a) \in R$. Then, by a similar argument, we obtain $[a] \subseteq [b]$. Consequently, $[a]=[b]$.

Now, assume $[a] = [b]$.

Then by part (i) of this proof that for each $a \in S$, we have $a \in [a]$. So, also, here $b \in [b]=[a]$; hence, $(a,b) \in R$.

Proof of (iii):

Here, we prove the equivalent contrapositive of the statement (iii), that is:

If $[a] \bigcap [b] \neq \emptyset$ then $[a] = [b]$.

if $[a] \bigcap [b] \neq \emptyset$ then there exists an element $x \in A$ with $x \in [a] \bigcap [b]$. Hence, $(a,x) \in R$ and $(b,x) \in R$. By symmetry, $(x,b) \in R$, and, by transitivity, $(a,b) \in R$. Consequently, by proof (ii), $[a] = [b]$.

The converse of the above theorem is also true. That is,

Theorem II:

Suppose $P = \{ A_{i}\}$ is a partition of set S. Then, there is an equivalence relation $\sim$ on S such that the set $S/\sim$ of equivalence classes is the same as the partition $P = \{ A_{i}\}$.

Specifically, for $a, b \in S$, the equivalence $\sim$ in Theorem I is defined by $a \sim b$ if a and b belong to the same cell in P.

Thus, we see that there is a one-one correspondence between the equivalence relations on a set S and the partitions of S.

Proof of Theorem II:

Let $a, b \in S$, define $a \sim b$ if a and b belong to the same cell $A_{k}$ in P. We need to show that $\sim$ is reflexive, symmetric and transitive.

(i) Let $a \in S$. Since P is a partition there exists some $A_{k}$ in P such that $a \in A_{k}$. Hence, $a \sim a$. Thus, $\sim$ is reflexive.

(ii) Symmetry follows from the fact that if $a, b \in A_{k}$, then $b, a \in A_{k}$.

(iii) Suppose $a \sim b$ and $b \sim c$. Then, $a, b \in A_{i}$ and $b, c \in A_{j}$. Therefore, $b \in A_{i} \bigcap A_{j}$. Since P is a partition, $A_{i} = A_{j}$. Thus, $a, c \in A_{i}$ and so $a \sim c$. Thus, $\sim$ is transitive.

Accordingly, $\sim$ is an equivalence relation on S.

Furthermore,

$[a] = \{ x: a \sim x\}$.

Thus, the equivalence classes under $\sim$ are the same as the cells in the partition P.

More later,

Nalin Pithwa.