On this Day:

In 1868, German mathematician Felix Hausdorff was born. He is considered a co-founder of general topology and made significant contributions to general and descriptive set theory, measure theory, functional analysis and algebra. In addition to his profession, he also worked as a philosophical writer and literary figure under the pseudonym Paul Mongré.

Felix Hausdorff – Early Years

Felix Hausdorff was born in Breslau, Kingdom of Prussion, today Wroclaw in Poland. Hausdorff’s father, the Jewish businessman Louis Hausdorff (1843-1896), moved with his young family to Leipzig in autumn 1870. His mother Hedwig (1848-1902) came from the widely ramified Jewish family Tietz. From 1878 onwards, Felix Hausdorff attended the Nicolai Grammar School in Leipzig, an institution which had an excellent reputation as a plantation of humanistic education.  In his graduation in 1887, he was the only one who reached the highest grade. From 1887 to 1891, Hausdorff studied mathematics and astronomy, mainly in his hometown Leipzig, interrupted by visiting semesters in Freiburg and Berlin. In the final semesters of his studies, Hausdorff closely followed Heinrich Bruns (1848-1919). Bruns was professor of astronomy and director of the observatory at the University of Leipzig. Hausdorff received his doctorate from him in 1891 with his thesis Zur Theorie der astronomischen Strahlenbrechung über die Refraktion des Lichtes in der Atmosphäre (On the theory of astronomical refraction on the refraction of light in the atmosphere). In 1895 earned his habilitation with a thesis on the extinction of light in the atmosphere. In the period between his doctorate and his habilitation, Hausdorff completed his one-year voluntary military service and worked for two years as a computer at the Leipzig observatory.

Literary Efforts

With his habilitation, Hausdorff became a private lecturer at the University of Leipzig and began an extensive teaching career in a wide range of mathematical fields. Besides teaching and research in mathematics, he pursued his literary and philosophical inclinations. The years 1897 to about 1904 mark the peak of his literary and philosophical work; during this period, 18 of the 22 writings published under pseudonyms appeared, including a volume of poetry, a play, a book critical of knowledge and a volume of aphorisms. His preferred pseudonym was Paul Mongré (à mon gré means: as I wish, as I like it)

Set Theory

In December 1901 Hausdorff was appointed extraordinary professor at the University of Leipzig. However, his main field of work soon became set theory, especially the theory of ordered sets. It was initially a philosophical interest, which led him to study Georg Cantor‘s work around 1897. Already in 1901 Hausdorff gave a lecture on set theory. This lecture already contained Hausdorff’s first discovery in set theory. In the summer semester of 1910, Hausdorff was appointed as a regular associate professor at the University of Bonn. In the summer of 1912 work also began on his opus magnum, the book Grundzüge der Mengenlehre (Basic Principles of Set Theory), which was published in 1914. Hausdorff was appointed full professor at the University of Greifswald for the summer term of 1913. This university was the smallest of the Prussian universities.

Theory of Ordered Sets

Hausdorff’s questions were the starting point for the theory of unattainable cardinal numbers. Hausdorff had already noticed that such numbers, if they exist, must be of “exorbitant size”.

National Socialist Regime

When the Nazi party came to power, Hausdorff thought that the university professor he was would not be worried. However, his mathematical work was denounced as “Jewish”, useless and “non-German”, and he lost his position in 1935. On 31 March 1935 Hausdorff was finally granted regular emeritus status after some back and forth. Those responsible at the time could not find a word of thanks for 40 years of successful work in the German higher education system. He continued to work tirelessly and, in addition to the extended new edition of his Set Theory, published seven papers on topology and descriptive set theory, all of which appeared in Polish journals: one in Studia Mathematica, the others in Fundamenta Mathematicae.

Final Years

In 1939, Hausdorff attempted in vain to obtain a research fellowship in the USA via the mathematician Richard Courant in order to be able to emigrate after all. Finally, in mid-1941, work began on deporting the Bonn Jews to the convent “Zur ewigen Anbetung” in Bonn-Endenich, from which the nuns had been expelled. From there they were later transported to the extermination camps in the East. After Felix Hausdorff, his wife and his wife’s sister Edith Pappenheim, who lived with them, had received the order to move to the Endenich camp in January 1942, they died together on 26 January 1942 after taking an overdose of Veronal (per http://scihi.org/felix-hausdorff/).

What is set theory?

Set Theory is a branch of mathematical logic where we learn sets and their properties. A set is a collection of objects or groups of objects. These objects are often called elements or members of a set. For example, a group of players in a cricket team is a set. 

Since the number of players in a cricket team could be only 11 at a time, thus we can say, this set is a finite set. Another example of a finite set is a set of English vowels. But there are many sets that have infinite members such as a set of natural numbers, a set of whole numbers, set of real numbers, set of imaginary numbers, etc. 

Set Theory Origin

Georg Cantor (1845-1918), a German mathematician, initiated the concept ‘Theory of sets’ or ‘Set Theory’. While working on “Problems on Trigonometric Series”,  he encountered sets that have become one of the most fundamental concepts in mathematics. Without understanding sets, it will be difficult to explain the other concepts such as relations, functions, sequences, probability, geometry, etc.

Definition of Sets

As we have already learned in the introduction, set is a well-defined collection of objects or people. Sets can be related to many real-life examples, such as the number of rivers in India, number of colours in a rainbow, etc. 

Representation of Sets

Sets can be represented in two ways:

  • Roster Form or Tabular form
  • Set Builder Form

Roster Form

In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }. 

Example: If set represents all the leap years between the year 1995 and 2015, then it would be described using Roster form as:

A = {1996,2000,2004,2008,2012}

Now, the elements inside the braces are written in ascending order. This could be descending order or any random order. As discussed before, the order doesn’t matter for a set represented in the Roster Form. 

Also, multiplicity is ignored while representing the sets. E.g. If 
L represents a set that contains all the letters in the word ADDRESS, the proper Roster form representation would be

L = {A,D,R,E,S } = {S,E,D,A,R}  

L≠ {A,D,D,R,E,S,S}

Set Builder Form

In set builder form, all the elements have a common property. This property is not applicable to the objects that do not belong to the set. 

Example: If set S has all the elements which are even prime numbers, it is represented as:

S = { x: x is an even prime number}

where ‘x’ is a symbolic representation that is used to describe the element.

‘:’ means ‘such that’

‘{}’ means ‘the set of all’

So, S = { x:x is an even prime number } is read as ‘the set of all x such that x is an even prime number’. The roster form for this set S would be S = 2. This set contains only one element. Such sets are called singleton/unit sets.

Another Example:

F = {p: p is a set of two-digit perfect square numbers}


F = {16, 25, 36, 49, 64, 81}

We can see, in the above example, 16 is a square of 4, 25 is square of 5, 36 is square of 6, 49 is square of 7, 64 is square of 8 and 81 is a square of 9}. 

Even though 4, 9, 121, etc., are also perfect squares, they are not elements of the set F, because it is limited to only two-digit perfect squares.

Types of Sets

The sets are further categorized into different types, based on elements or types of elements. These different types of sets in basic set theory are:

  • Finite set: The number of elements is finite
  • Infinite set: The number of elements are infinite
  • Empty set: It has no elements
  • Singleton set: It has one only element
  • Equal set: Two sets are equal if they have same elements
  • Equivalent set: Two sets are equivalent if they have same number of elements
  • Power set: A set of every possible subset.
  • Universal set: Any set that contains all the sets under consideration.
  • Subset: When all the elements of set A belong to set B, then A is subset of B (per https://byjus.com/maths/basics-set-theory/)

First, a Story:

A piece of string walked into a small town on a hot, dusty day. He was thirsty, so he sauntered into the first establishment he
encountered and asked the waiter for a glass of water.

“Sorry”, said the waiter, “we don’t serve strings here.”

Discouraged, the string walked out.  A little further down the street, he met a mathematician.

“You look hot,” said the mathematician.  “Why don’t you go into that cafe and get a drink of water?”

“I tried that,” said the string, “but the waiter wouldn’t serve me anything because I’m just a string.”

“No problem” said the mathematician.  “I’ll fix you up.”  He grabbed the string, tied him in a bowline and frayed his ends.  “Now try it.”

The string slipped back into the cafe and asked the waiter for a glass of water.  “Hey,” said the waiter, “aren’t you the piece of
string that was just in here?”

“Nope,” retorted the string, “I’m a frayed knot.”

Second, a Song:

Math isn’t hard, it’s a language, per Randy Palisoc speaking at TEDxManhattanBeach.

This talk was given at a local TEDx event, produced independently of the TED Conferences. Is 26% proficiency in math acceptable to you?  That’s the question teacher and Synergy Academies Founder Dr. Randy Palisoc asks the TEDxManhattanBeach audience.  With only 26% of U.S. twelfth graders proficient in mathematics, Randy shares his solution: teach math as a language. Putting words back into math lessons enables even the youngest school-age minds to grasp complex concepts, such as fractions, that are traditionally thought of as abstract and difficult to understand.  In his stunningly simple and effective approach, math no longer creates problems for kids but solves them.

Randy Palisoc is a passionate educator, known for making math easy.  A founder of the five-time national award winning Synergy Academies, Randy is proud to identify Synergy as an “in spite of” school.  In spite of the fact that its students were disadvantaged and in spite of the fact that it did not have its own facility and had to pack up its classrooms every single week, his school was the first and only elementary school in South Los Angeles to ever win the National Blue Ribbon Award, and was named the #1 Urban Elementary School in America in 2013.

Seeking a broader impact, Randy left Synergy this year to start Ironbox Education.  One of the ways he believes we can dramatically improve America’s schools is by doing a better job teaching math to our students in a way that makes sense to them.  Randy believes implementing a few changes in the way we approach math will give students the confidence and thinking skills to be more competitive in a global economy.

Randy’s Favorite TED Talk:  “My Invention That Made Peace With Lions”: Richard Turere

The “Game Changer” he most admires is Steve Jobs, because he had the ability to envision things we did not even know we needed or wanted.

About TEDx, x = independently organized event In the spirit of ideas worth spreading, TEDx is a program of local, self-organized events that bring people together to share a TED-like experience. At a TEDx event, TEDTalks video and live speakers combine to spark deep discussion and connection in a small group. These local, self-organized events are branded TEDx, where x = independently organized TED event. The TED Conference provides general guidance for the TEDx program, but individual TEDx events are self-organized.*  (*Subject to certain rules and regulations) per YouTube.com).

Here is Randy Palisoc on “Math isn’t hard, it’s a language”. I hope you enjoy this!


Thought for the Day:

“The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe.” – Karl Pearson

Have a great day!

Dave & Colleen

© 2021 David J. Bilinsky and Colleen E. Bilinsky

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