# Author: Nalin Pithwa

# How to sew a two holed torus

# Wisdom of Abraham A. Fraenkel

Before looking at a proof, the student should realize what is to be proved, why a proof is necessary, and what is the main difficulty to be overcome.

# Difference between identity and equality

A nice way to illustrate this is the following conversation in Lewis Caroll’s Alice in Wonderland:

In a discussion between Alice and the Pigeon, the latter accuses her of being a serpent. In spite of her indignation, Alice has to confess that “little girls eat as many eggs as serpents do.” But, the Pigeon calls “anything with long necks and eating many eggs” as “a serpent.”

In other words, two objects may have same property and are equal in that sense. But, the objects might be different.

So, that brings to mind the next question: what is the difference between equal and equivalence. In my opinion, equivalence measures equality up to a certain property. For example, congruence modulo relation in numbers. Or, congruent triangles in Euclid’s geometry.

Regards,

Nalin Pithwa

# Exercises: Set theoretic construction of real numbers

Exercises:

- If p is a prime number, show that is irrational.
- Show that
- Show that if , then the number is an irrational number.
- Show that if and are two Cauchy sequences, then so are and .
- If A and B are non empty subsets of that is bounded below. Then, if and only if (i) for all and (ii) for any , .

Reference: A Second Course in Analysis by M Ram Murty, Hindustan Book Agency.

Regards,

Nalin Pithwa

# Set theoretic construction of real numbers

Continued from previous blog. Same reference: A Second course in analysis by M Ram Murty, Hindustan Book Agency.

Section 1.3

The rational number is insufficient to “measure” all the lengths that arise in the “real world.” This was discovered by the ancient school of Pythagoras which viewed the world through a strange mix of mathematics and mysticism. The aphorism “all is number” seems to have been the underlying mantra of the Pythagoreans. In our modern digital world, this mantra imposes the universality more than in any earlier age.

Pythagoras seems to have been a contemporaryof the Buddha in India, of Confucius and Lao-Tzu in China. He seems to have travelled widely in Egypt, Babylon and India. The Pythagoreans believed in reincarnation and this was wedded to their strict adherence to vegetarianism for by eating meat they might be eating a friend!They are credited to the discovery of both the words philosophy and mathematics. The word “philosophy” literally means the love of wisdom and the word “mathematics” means “that which is learned.” Philosophy then, for the Pythagoreans was seen as a byproduct of mathematics.

The celebrated theorem of Pythagoras dealing with the length of the hypotenuse of a right angled triangle was known to earlier cultures stretching far back in time to Babylon, China and India. It is possible that Pythagoras learned of this theorem during his extensive peregrinations. But the Pythagoreans could see the mysticism inherent in the theorem, for the rule applied to all right angled triangles and there was an element of universality in it and in mathematics in general.

But when the Pythagoreans declared “all is number” they were referring to whole numbers and it is accurate to say that the concept of number had not been clearly articulated at that time. With the underlying idea that whole numbers govern the universe, their theory of ratio and proportion was in for a rude shock. Legend has it that the Pythagorean who discovered that is an irrational number was drowned !

The irrationality of represents a major episode in the history of mathematics and we owe this to the unfortunate Pythagorean who lost his life by discovering it. The proof is by contradiction and invokes the rudimentary arithmetical idea of a prime number, a concept first enunciated in Euclid’s Elements. A natural number is called a prime number if it has no proper divisors. Thus, the sequence of prime numbers begins as (The number one is not included in the list of prime numberss. It it called a unit, a concept which generalizes to rings.) Two natural numbers a and b are said to be coprime if they have no common prime factor. When we write a fraction a/b in “lowest termss”, we are writing it such that a and b are coprime. We sometimes write to indicate that a and b are coprime. This is not to be confused with the concept of an ordered pair introduced earlier.

The irrationality of is established by using the method of contradiction. It is instructive to see all the details. If is a rational number, then we can write

such that

Upon squaring both sides, we see

which implies that a is even since the left hand side of the equation is patently an even number. Writing , we now get

and so b is even. But this contradicts the coprimality of a and b.

This was the theorem that led to the demise of the heroic Pythagorean for it violated the aphorism “all is number” because the ancients viewed numbers as only whole numbers. They also allowed for rational numbers since they can be represented as ratio and proportion of whole quantities. This innocuous theorem opens the door for the discovery of real numbers. For it signals the lack of the “least upper bound property” in the realm of rational numbers.

The student should keep in mind that so far, we have only constructed the universe of rational numbers and thus any further definitions must be given only in terms of rational numbers. A set A of rational numbers is said to be bounded if there is a rational number M such that for all . An upper bound for A is any rational number u such that for all . A lower bound for A is a rational number l such that for all . A rational number v is called a least upper bound for A if for every upper bound u of A. A rational number m is called a greatest lower bound for A if for every lower bound l of A. Thus, a set A is bounded if and only if it has both an upper bound and a lower bound.

Several questions now arise: given a set A of rational numbers, does it always have a least upper bound? Does it have a greatest lower bound? If so, are they unique?

The theorem discovered by the Pythagoreans can be explained as follows. Nascent in this example are many of the fundamental properties we would expect of the real numbers. Essential to the understanding is the familiar fact that the sequence of rational numbers gets smaller and smaller and tends to zero.

If we consider the set

then A has no least upper bound. Indeed if u is a least upper bound, for any . By definition, u is a rational number and so cannot first equal . If , then we can choose a natural number N such that

so that

contradicting that u is an upper bound for the set A. If , then we can similarly find N such that

,

which again contradicts that u is the least upper bound for A. Thus, A does not have a least upper bound in the realm of rational numbers.

Digress: We digress a bit here from Dr. M Ram Murty’s above text and reproduce a nice “little proof”‘ from Rudin’s Analysis text illustrating that the rational number line has gaps or holes in it..

****

As we know there exists no rational solution to . We now examine this situation a little more closely. Let A be the set of all positive rationals p such that and let B consist of all positive rationals p such that . We shall show that A contains no largest rational number and B contains no smallest rational number.

More explicitly, for every we can find a rational q in A such that , and for every p in B, we can find a rational q in B such that .

To do this, we associate with each rational number the number q such that:

…call this I.

Thus, ….call this II.

If p is in A, then , (1) above shows that , and II shows that . Thus, q is in A.

If p is in B, then , (I) shows that , and II shows that . Thus, q is in B.

Conclusion: the purpose of the above discussion is to show that the rational number system has gaps, in spite of the fact that in between any two rational numbers, there is another: if , and if , then . The real number system fills these gaps. This is the principle reason for the fundamental role which it plays in analysis.

***

Two ways of constructing the real numbers were proposed independently by Richard Dedekind (1831-1916) and Augustin Cauchy (1789-1857). Each method has its virtues. The method of Dedekind, using what are called Dedekind cuts is closer to the axiomatic foundations we have been discussing and is suggested by the example discussing above. The method of Cauchy using what are now called Cauchy sequences has a wider applicability. In our example above, an essential role is played by the ordering of the rational numbers. The method of Dedekind cuts uses this idea in a fundamental way. It coincides with our intuitive and visual idea of the real numbers as being “points on the number line stretching from minus infinity to plus infinity.”

A Dedekind cut in Q is a partition of Q into two sets A and B with the following properties:

- A and B are both non-empty.
- A is “closed downwards”, that is, if and , then ;
- B is “closed upwards”, that is, if and , then .
- A contains no greatest element, that is, there is no such that for all .

The set R of real numbers is then defined as the set of all Dedekind cuts.

For example, the irrational number is defined by the Dedekind cut:

, and , and .

For example, the number zero is represented by with A being the set of all non-zero negative rational numbers and B being the set of all non-zero positive rational numbers. As the Dedekind cut is uniquely defined by A, with B being the complement of A in the set of rationals, we may as well associate A to each real number, or even think of A as the real number. One would then have to define how one would work with this definition from the perspective of addition, multiplication and so on. This is not difficult to do. For instance, we have

, and .

Addition is simple enough:

Multiplication is defined (rather cumbersomely) by setting to be

if

, if , and

, if ,

, if

The absolute value function is then

, if ; , if ; , if .

One can prove that all the usual operations with numbers satisfy the expected properties of commutativity, associativity, and distributivity.

By contrast, the method of Cauchy sequences begins with a definition of convergence. In some ways, it is motivated by our discussion of earlier. If A is the Dedekind cut representing , then the least upper bound of A “is” . Thus, the number can be thought of as a limit of a sequence of rational numbers. But there can be many such sequences and so one needs a more formal approach. The underlying observation is that a limit of a sequence of rational numbers need not be rational.

This is perhaps best illustrated by Euler’s series for e. Students of calculus are familiar with

and the right hand side can be thought of as a limit of a sequence of rational numbers

The irrationality of e is easily proved by the method of contradiction. Suppose that for two coprime natural numbers a and b. Then,

…call this 1.1

and the second term on the right hand side is strictly less than the geometric series

which is less than or equal to 1. But the left hand side of 1.1 is a positive integer, the first term on the right hand side of 1.1 is a positive integer and the second term is not an integer, which is a contradiction.

What this example shows is that the “limit” of the sequence of partial sums (each of which is a rational number) is the irrational number e.

The reader is familiar with the usual notion of convergence. We say a sequence of real numbers converges to L if given there exists N such that

for all

One may want to view real numbers as “limits of sequences of rational numbers.”But in our formal construction of numbers, several difficulties arise with this definition. The first is that as we have so far only constructed the rational numbers, we need to take the ‘s to be all rational numbers. Secondly, the limit L need not be a rational number as the example with e shows. This motivates the formal definition of a Cauchy sequence of rational numbers.

A sequence in is called a Cauchy sequence if given , there is an N such that

for all

Now it is easy to see that any convergent sequence is a Cauchy sequence. For reference, we record this below:

Theorem 1.2: Any convergent sequence is a Cauchy sequence.

Proof of 1.2: Suppose that converges to L. Then, choosing N such that

for all

we have for all , by the triangle inequality

.

QED.

Later, we will show that every Cauchy sequence converges. Thus, a sequence is Cauchy if and only if it is convergent and the two notions are equivalent. But the advantage of the notion of a Cauchy sequence is that the limit value L is not mentioned in the definition.

The construction of real numbers is now brought about by defining an equivalence relation on the set of all Cauchy sequences of rational numbers. Since we have constructed rational numbers, we are now allowed to construct the set of all Cauchy sequences of rational numbers. On this set, we say two Cauchy sequences and are equivalent if for any , there exists a natural number N such that

for all .

This is easily seen to define an equivalence relation on the set of all Cauchy sequences of rational numbers. The set of real numbers is then defined as the set of all equivalence classes. We will denote the equivalence class of the sequence by .

We leave as an exercise to verify that if and are two Cauchy sequences, then so are and . Thus, the usual operations of addition and multiplication are easily defined as:

One needs to verify that these are well-defined by checking that the definition does not depend on the choice of representative of the equivalence class.

The order relation of the real numbers can also be defined. Thus, viewing real numbers as equivalence classes or Cauchy sequences of rational numbers, we define if there is an N such that for all . We also write if either or . Again, these order relations have the expected properties.

Finally, the absolute value function can also be defined. We set

$latex|[\{ q_{n}\}]| = [\{ |q_{n}|\}]$

and the reader can easily verify that indeed the right hand side is a Cauchy sequence if is Cauchy.

The absolute value has the usual properties, the most notable being the triangle inequality

Having given this formal definition of the real numbers using Cauchy sequences, it is convenient to drop the sequence notation and just represent the real numbers as single letters. Thus, if a, b are real numbers, the triangle inequality is the familiar:

It is now convenient to visualize the set of real numbers in the usual way as points of the line stretching from to .

By the very construction, the set of real numbers has the property of completeness in that every Cauchy sequence converges. The reader will also observe that the construction depended only on the property of the absolute value viewed as a metric to measure the distance between two rational numbers. This signals a wider applicability of the method of Cauchy sequences valid more generally in metric spaces which we discuss later.

With the construction of the real numbers, we have restored the least upper bound property and can now define that a sequence of numbers (rational or real) converges to a number if given any , there is an N such that

for all .

A non-empty subset A of is said to be bounded below if there is a number such that for all . Any such number s is called a lower bound for A. Analogously, A is said to be bounded above if there is an upper bound for A. The set A is said to be bounded if it is both bounded below and bounded above and unbounded if it is not bounded.

The construction of the real numbers using the Dedekind cuts shows that every non-empty subset of that is bounded above has a least upper bound denoted by . The least upper bound is also called the supremum. Every non-empty subset which is bounded below has a greatest lower bound denoted by . The greatest lower bound is also called the infimum. It is clear that if A and B are non-empty subsets of such that , then (see exercises). The following theorem gives a convenient characterization of the supremum.

Theorem 1.3 Let A be a non-empty subset of that is bounded above. Then if and only if

(i) for all , and

(ii) for any , .

Proof :

Let . Then s is an upper bound for A so that (i) holds. Now if (ii) were false, there would be an such that . But then would also be an upper bound for A, a contradiction. Conversely, suppose thatt (i) and (ii) hold. Then (i) implies s is an upper bound for A. If s were not the least upper bound for A, there would bea such that for all. Taking in (ii) we get that . But so that which is a contradiction since for all .

QED.

Theorem 1.3 can also be stated as follows. If A is a non-empty subset of that is bounded above and, then either or for all . A similar theorem can be written for the infimum of a set.

Cheers,

Nalin Pithwa

# Constructing numbers from sets

Continued from previous blog: A fast review of set theory; same reference: A Second Course in Analysis by M Ram Murty, Hindustan Book Agency.

Mathematicians and philosophers of the nineteenth century pondered deeply into the nature of a number. The question of “what is a number?” is not a simple one. But since mathematicians decided to give foundations of mathematics using the axiomatic method and sets as the basic building blocks, we are led to define numbers using sets. We follow Richard Dedekind (1831-1916) and Giuseppe Peano (1858-1932) in the following construction. It was as late as 1888 and 1889 when this construction was described in two papers written independently by Dedekind and Peano.

We construct a sequence of sets to represent the natural numbers. As noted earlier, zero is represented by the empty set. We have already described the construction of the natural numbers using the empty set. For each natural number n, the successor of n is denoted by (and sometimes as ) and defined as

Thus, each natural number is a set with n elements, namely,

We designate the set of natural numbers by the symbol . (It is a matter of personal convenience whether to include zero as a natural number or not. In this discussion, zero is a natural number. In other settings, it may not be. There is no universal convention regarding this and the student is expected to understand depending on the context. Some authors use the term “whole numbers” to indicate that zero is included in the discussion.)

The arithmetic operations on are now defined recursively. Addition is defined as a function from to :

where is defined recursively by and . A similar definition is given for multiplication x by defining and

We also define as simply mn which is the familiar symbology.

An equivalence relation on a set S is a subset R of satisfying:

- (reflexive axiom) .
- (symmetry axiom) .
- (transitive axiom) and implies .

The notion of an equivalence relation is an abstaction of our concept of equality, or at least what we implicitly expect of the notion of equality. It is more suggestive to write the equivalence relationn, not as a subset of as indicated above, but rather more symbolically as that our axioms become:

- (reflexive) , .
- (symmetry) if and only if .
- (transitive) and implies .

Equivalence relations play a fundamental role in all of mathematics. They allow us to understand aspects of sets by grouping them using certain properties.

To construct negative integers, we define an equivalence relation on . We write

.

Intuitively, we think of as so that it becomes evident that our definition is now in terms of concepts that have been defined earlier. This is very similar to how the ancients worked with negative numberss that appeared in an equation. They usually moved them to the other side so that the equation became an equation of non-negative numbers. However, with our set theoretic definition, we have reached a more fundamental and higher level of abstraction. Thus, with our equivalence relation above on the natural numbers, we define the set of integers as the set of equivalence classes of such ordered pairs. It is now easy to see that the following lemma holds:

Lemma 1.1 If is an ordered pair of non-negative integers, then exactly one of the following statements holds:

(a) is equivalent to for a unique non-negative integer m;

(b) is equivalent to for a unique non-negative integer m;

(c) is equivalent to .

Sometimes, we denote by the equivalence class of . With this lemma in place, we now denote by m the set of pairs of non-negative integers equivalent to ; by -m the set of pairs equivalent to and by 0 the set of pairs equivalent to . We denote these equivalence classes by .This gives us set theoretic construction of the set of integers.

We can define the operations of addition and multiplication by setting:

This latter definition is best understood if we recall that the symbol represents j-k so that the left hand side of the above equation is

One needs to check that these definitions are “well-defined” in the sense that they are independent of the representatives chosen for the equivalence class. This can be done as exercises.

In this way, we have now extended the notion of addition and multiplication from the set of natural numbers to the set of integers. Subtraction of integers can be defined by

where -1 represents the equivalence class . All of these definitions correspond to our usual notion of addition, subtraction and multiplication. Their virtue lies in their pure set-theoretic formulation.

We can also order the set of integers in the usual way. Thus,

and

.

This corresponds to our usual notion of “less than” and “less than or equal to”.

Finally, we can define the absolute value on the set of integers by setting

, if

, if

, if

We can now construct the rational numbers from the set of integers. We do this by defining an equivalence relation on the set by stating that two pairs and are equivalent if and only if . Intuitively, we think of as representing the “fraction” and examining what we would mean by by reducing it to notions already defined. The set of rational numbers is then defined as the set of such equivalence classes.

The expected operations of addition and multiplication are now evident:

Again, these definitions are easily verified to be well-defined. Finally, we can now define “division”.If with , we define:

These operations satisfy the familiar laws of associativity, commutativity and distributivity. Subtraction of rational numbers then can be written as :

The ordering of rational numbers can also be written as:

.

These definitions agree with out usual notions of ordering of the rational numbers.

Finally, the definition of absolute value can be extended as:

if

if

.

Again, our familiar properties of the absolute value of rational numbers hold. With this foundational construction in place, we can conveniently represent the equivalence class of as simply the fraction j/k and continue to work with these numbers as we were hopefully taught from childhood.

In the next sections/blogs we construct the real numbers from this axiomatic framework.

Exercises:

Hint (generic): keep the meaning of the symbols in mind and meaning of equivalence relations and equivalence classes. Also note that our basic object is a class and a set is a member of a class.

- let be two elements of . Show that the addition:

is well-defined. That is, prove that for any and , we have that is equivalent to .

2. For , prove the distributive law: .

3. Show that the relations < and on have the following properties:

(a) for all

(b) for all

(c) if and only if

(d) for all

(e) if and only if .

(f) for all .

(g) for all

(h) for if and only if

(i) for all

(j) where if and only if .

Cheers,

Nalin Pithwa.

# Boarbarktree

# Axioms of set theory: fast review

Reference: A Second Course in Analysis, M Ram Murty, Hindustan Book Agency.

Chapter 1 Background

Axioms of Set Theory

The concept of number lies at the heart of mathematics. The subject of mathematical analysis, in the formal sense of the word, began when mathematicians initiated the exploration of infinity as a precise mathematical idea. This perhaps is the greatest event of the nineteenth century, on par with the discovery of zero as a mathematical idea. Both zero and infinity are now essential to the understanding of limits and the related processes of real and complex analysis.

Because these two ideas were poorly understood, mathematicians in the nineteenth century introduced the axiomatic method with the theory of sets as the foundation for all mathematics. In this approach, Georg Cantor (1845-1918) was definitely the leading figure. Sadly, his work received considerable opposition at first, but is now universally accepted. It is not an exaggeration to say that research into the foundations of mathematics has given us a larger understanding of the nature of mathematical logic, both its strengths and limitations.

Set theory is the standard foundation of all mathematics. Mathematical proofs use set-theoretic ideas in one form or another. Below, we give a quick overview of the axioms of set theory so that the student is introduced to the relevant vocabulary. One need not become overly preoccupied with these formalities for otherwise a serious study of these axioms will take us into the realm of mathematical logic which is not the province of this short survey. Rather, we indicate the main themes, very much similar to a quick sight-seeing tour of an ancient city. There will undoubtedly be many side streets to explore and since time is of the essence, we need to be content with a one-hour tour highlighting the significant monuments. For a gentle introduction to mathematical logic, we refer the reader to the following text: Hilbert’s Tenth Problem, An introduction to logic, number theory and computing, Student Mathematical Library, Volume 88, American Mathematical Society, Providence, Rhode Island, 2019.

The basic objects of set theory are classes, some of which are sets. An informal definition of a set is an unordered collection of objects. The basic relationship between them is membership indicated by the symbol . Thus, $x \in A$ means that x is a member of A. Its negation, x is not a member of A is written as $x \notin A$. Already, with the notion of a set and the relation of membership, we can easily run into logical contradictions if we do not formulate proper axioms for the construction of sets. The famous “Russell paradox” (named after the British philosopher and mathematician Bertrand Russell (1872-1970)) shows that the “set of all sets” is not a set. Indeed, if it were a set, it would be a proper member of itself by definition, which is a contradiction. The paradox is often posed as the following riddle: In the town of Seville lives a barber who shaves everyone who does not shave himself. Does the barber shave himself? The contradiction arises from self-reference and so it was quickly realized that to avoid logical contradictions in mathematics, one must formulate precise axioms for the construction of sets. In a formal system of axiomatic set theory, the notions of “class” and “” are undefined terms. There are several well-developed axiomatic systems such as the Zermelo-Fraenkel system and the von Neumann-Godel-Bernays system. Our treatment below follows the latter.

The underlying idea of any these systems is that we construct new sets from old sets in a systematic manner, with the existence of the empty set as a basic axiom. This is how we avoid the Russell paradox.

The words “and” and “not’ are used as propositional connectives. We usually write these words out and do not use symbols for them, though some logicians do. To say that “Proposition P implies Proposition Q” we write ““. If $P \Longrightarrow Q$ and $Q \Longrightarrow P$, we write . The usual logical quantifiers are : which means “for all” and which means “there exists”. The first one is sometimes called the universal quantifier and the second one as the existential quantifier. The statement, “For all x, the proposition F(x) holds” is denoted $\forall (F(x))$ with the parentheses used to designate the proposition. Such a statement is called an existential proposition.Using class variables like x, y, A,B and so on, together with the basic relation of membership $\epsilon$, quantifiers, connectives and parentheses, we can construct “well-formed” formulas. If variables are quantified, they are called “bound” and should only appear after the quantifier. Variables without quantifiers are called “free”. For example,

is well-formed in which x and y are bound variables and z is free, whereas

is not well-formed since z precedes its quantifier.

As noted earlier, it is not our goal here to develop axiomatic set theory. On the other hand, the mature mathematician cannot ignore the need for foundations and must have some idea of the basic axioms and notation. So we content ourselves with a cursory overview of the basic axioms and set the stage for both notations and structure of logical proofs.

The essential idea of set theory is that sets are built up in a progressive hierarchy and there are precise rules as to how to form new sets from old sets. We begin with the axiom of extensionality which allows us to determine when two sets are equal.

Axiom I : (extensionality) For any classes A and B, if and only if

.

This axiom can be taken as the definition of equality.

Axiom 2: (existence of the empty set). There is a set such that

The axiom of extensionality implies that the empty set is unique.

Axiom 3: (unordered pairs) For all sets x, y there is a set such that

If we obtain the singleton , if and only if . Again this sets are unique by extensionality.

As defined earlier, a variable is free if there are no quantifiers defining it. Otherwise, the variable is said to be bounded. This concept isused in the next axiom and allows us to form new sets from old using well-formed formulas. In fact, the next axiom is more appropriately called an axiom schema or a collection of axioms.

Axiom 4: (selection): Given any well-formed formula containing one free variable x, there is a class A suchthat for any set x, we have $x \in A$ if and only if holds.

We often write

and read this as “A is the class of all (sets) x such that holds.”

The distinction between sets and classes in the last axiom is important as noted earlier. Without it, we fall into”Russell’s paradox.” Indeed, let . If R were a set, then implies and if , then . a contradiction in both cases. Technically, R is not a set but a proper class containing all sets which we call the “universe” and the theory contains no logical contradictions. The previous axioms do allow us to construct new sets from old ones in a systematic manner. Thus, a set may consist of other sets that have already been well-defined.

Axiom 5 (union): For any set x, there is a set such that if and only if for some .

We sometimes write for in the last axiom. Thus,

.

For any well-formed formula with a free variable x, means

.

The relation of A being a subset of B is defined by the symbol:

and we write this as , or equivalently .

Axiom 6(power set axiom); For any set x, there is a set such that if and only if .

The reader should understand that the symbol is pure symbology for the set of all subsets of x and that no exponentiation is meant since x is a set. It will always be clear from the context which meaning is indicated.

For any two classes A and B, we define the intersection as

.

If A is any set, then is a set for any class B by Axiom 6 and the definition of a set. If x is a non-empty set, its intersection is defined by

.

This is easily seen to be a set. We can also write as .

We can use set theory to define an ordered pair as the set of the form .

Indeed, for any ordered pair Q, its first member is the unique x such that and its second member is the unique y such that , where x is the first member.

A relation is any set of ordered pairs. For any relation E, we define the inverse relation by

.

From the definition of an ordered pair, we are led to the definition of a function as a set f of ordered pairs such that and implies . We write to mean for any given function f. Thus, a function is a special kind of relation. In other words, all functions are relations, but not all relations are functions.

If f is a function, is not necessarily a function. A function f is called one-to-one if and only if is a function.

This now allows us to construct the Cartesian product of two sets X and Y denoted , which consists of ordered pairs with and .

With the axioms listed so far, we cannot prove the existence of an infinite set. The next axiom allows us to do this along with our earlier axioms.

Axiom 7 (axiom of infinity) There is a set such that for all , .

This axiom allows us to construct the natural numbers using the empty set because we now define inductively the sequence of sets

, , ,

and define the non-negative integers using the “zero-symbol” 0 to designate the empty set and define

, , ,

We could also have stated Axiom 7 in logical notation as such that and ,

To prevent from being equal to x, and in general to keep order among sets and prevent unwanted closed cycles for the membership relation, we need a further axiom:

Axiom 8: (axiom of regularity) For any set , . This allows us to prove :

Theorem 1.1: for all x and y, and if , then .

Proof 1.1: by Axiom 8, since . If , then clearly which implies , a contradiction. This proves the first half of our assertion. For the second half, if , then by Axiom 8, the set has a member disjoint from it. This member cannot be x for otherwise , a contradiction by the first half of the theorem. If the member is y, then is also a contradiction since . QED.

Exercises:

- Recall that an ordered pair can be defined as the set . Show that if and only if and .

Solution 1: This is clear by definition of equality and definition of ordered pair.

2. Define the ordered triple to be the ordered pair where the ordered pair is defined a usual. Show that

if and only if .

Solution: If , then by definition of triple,

, which in turn by equality of ordered pairs means that

and that is that is and .

3. If , prove that .

Solution: need not mean unless .

4. Which of the following are functions and why?

4. (a) . yes.

4. (b) . No.

4 (c). . yes.

4 (d) . No.

4 (e) . yes.

5. For any relation V (that is, any set of ordered pairs), define the domain of V as

and the range of V as

Find the domain and range for each relation in the previous question whether or not it is a function.

4a. Domain is , and same range.

4b. Domain is and range is .

4c. Domain is , range is .

4d. Domain is and range is /

Cheers,

Nalin Pithwa

# An example: continuous map which is not homeomorphic

Reference: Topology by Hocking and Young. Dover Publications.

Let us present an example of a continuous mapping (one-to-one) which is not a homeomorphism:

Let S be the set of all non-negative real numbers with their usual metric topology, and let T be the unit circle in its metric topology. Consider $f: S \rightarrow T$. For each x in S, let $f(x) = (1, \frac{2\pi x^{2}}{(1+x^{2})})$, a point in polar coordinates on T.

It is easily shown that f is continuous and one-to-one.

But the set of all x in S such that $x<1$ is open in S while its image is not open in T. Hence, f is not interior. (meaning that: A transformation $f: S \rightarrow T$ of the space S into the space T is said to be interior if f is continuous and and if the image of every open subset of S is open in T) , and is not a homeomorphism (because of the following theorem: A necessary and sufficient condition that the one-to-one continuous map $f: S \rightarrow T$ of the space S onto the space S be a homeomorphism is that f be interior).

Regards,

Nalin Pithwa.