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Numbers and Their Application - Lesson 1

All About Sets

Lesson Overview

Sets, Elements, and Subsets

One dictionary has, among the many definitions for set, the following: a number of things naturally connected by location, formation, or order in time.

Although set holds the record for words with the most dictionary definitions, there are terms mathematicians choose to leave undefined, or actually, defined by usage. Set, element, member, and subset are four such terms which will be discussed in today's lesson. Today's activity will also explore the concept.

Each item inside a set is termed an element.

The brace symbols { and } are used to enclose the elements in a set.

Each element is a member of the set (or belongs to the set).

The symbol for membership is [is in]. It can be read "is an element of" and looks quite similar to the Greek letter epsilon ([epsilon]).

A subset is a portion of a set.

The symbol for subset is [subset symbol]. Some books will allow and use it reversed—we will not.

A superset is a set that includes other sets.

For example: If A B, then A is a subset of B and B is a superset of A.

A subset might have no members, in which case it is termed the null set or empty set.

The empty set is denoted either by {} or by [empty set symbol], a Norwegian letter. The null set is a subset of every set.

Note: a common mistake is to use {} to denote the null set. This is actually a set with one element and that element is the null set. Since some people slash their zeroes, it is safest when handwriting to always use the notation {} to denote the empty or null set.

A singleton is a set with only one element.

A subset might contain every member of the original set.
In this case it is termed an improper subset.

A proper subset does not contain every member of the original set.

Sets may be finite, {1, 2, 3,..., 10}, or infinite, {1, 2, 3,...}. The cardinality of a set A, n(A), is how many elements are in the set. The symbol ... called ellipses means to continue in the indicated pattern. There are 2n subsets of any set, where n is the set's cardinality—check it out for n=3!

The power set of a set is the complete set of subsets of the set.

In this class we will consider only safe sets, that is, any set we consider should be well-defined. There should be no ambiguity as to whether or not an element belongs to a set. That is why we will avoid things like the village barber who shaves everyone in the village that does not shave himself. This results in a contradiction as to whether or not he shaves himself. Also consider Russell's Paradox: Form the set of sets that are not members of themselves. It is both true and false that this set must contain itself. These are examples of ill-defined sets.

Sometimes, instead of listing elements in a set, we use set builder notation: {x| x is a letter in the word "mathematics"}. The symbol | can be read as "such that." Sometimes the symbol is reserved to mean proper subset and the symbol is used to allow the inclusion of the improper subset. Compare this with the use of < and to exclude or include an endpoint. We will make no such distinction. A set may contain the same elements as another set. Such sets are equal or identical sets— element order is unimportant. A = B where A = {m,o,r,e} and B = {r,o,m,e}, in general A=B if A B and B A. Sets may be termed equivalent if they have the same cardinality. If they are equivalent, a one-to-one correspondence can be established between their elements.

The universal set is chosen arbitrarily, but must be large enough to include all elements of all sets under discussion.

Complementary set, A', is a set that contains all the elements of the universal set that are not included in A. The symbol ' can be read "prime."

For example: if U={0, 1, 2, 3, 4, 5, 6,...} and A={0, 2, 4, 6, ...}, then A'={1, 3, 5, ...}.

Such paradoxes as those mentioned above, particularily involving infinities (discussed in the next lesson), were well known by the ancient Greeks. During the 19th century, mathematicians were able to tame such paradoxes and about the turn of the 20th century Whitehead and Russell started an ambitious project to carefully codify mathematics. Set theory was developed about this time and serves to unify the many branches of mathematics. Although in 1931 Kurt Gödel showed this approach to be fatally flawed, it is still a good way to explore areas of mathematics such as: arithmetic, number theory, [abstract] algebra, geometry, probability, etc.

Geometry has a long history of such systematic study. The ancient Greek Euclid similarily codified the mathematics of his time into 13 books called The Elements. Although these books were not limited to Geometry, that is what they are best known for. In fact, up until about my grandfather's day, The Elements was the textbook of choice for the study of Geometry! The Elements carefully separated the assumptions and definitions from what was to be proved. The concept of proof dates back another couple hundred years to the ancient Greek Pythagoras and his school, the Pythagorean School.

Intersection and Union

Once we have created the concept of a set, we can manipulate sets in useful ways termed set operations. Consider the following sets: animals, birds, and white things. Some animals are white: polar bears, mountain goats, big horn sheep, for example. Some birds are white: dove, stork, sea gulls. Some white things are not birds or animal (but birds are animals!): snow, milk, wedding gowns (usually).

The intersection of sets are those elements which belong to all intersected sets.

Although we usually intersect only two sets, the definition above is general. The symbol for intersection is .

The union of sets are those elements which belong to any set in the union.

Again, although we usually form the union of only two sets, the definition above is general. The symbol for union is .

For the example given above, we can see that:
{white things}{birds} = white birds
{white animals}{birds} = white animals and all birds
{white birds}{white animals}{animals}

Another name for intersection is conjuction. This comes from the fact that an element must be a member of set A and set B to be a member of A B. Another name for union is disjunction. This comes from the fact that an element must be a member of set A or set B to be a member of AB. Conjunction and disjunction are grammar terms and date back to when Latin was widely used.

I should note the very mathematical use of the word or in the sentence above. Common usage now of the word or means one or the other, but not both (excludes both). Mathematicians and computer scientists on the other hand mean one or the other, possibly both (including both). This ambiguity can cause all kinds of problems! Mathematicians term the former exclusive or (EOR or XOR) and the latter inclusive or. We will see ands & ors again in numbers lesson 6 on truth tables.

Pictures of Sets (Venn Diagrams)

John Venn (1834-1923) extended the use of diagrams first developed by Leonhard Euler (1707-1783), the great swiss mathematician, to give pictures of sets. Venn diagrams are often used to visualize set operations.

A superset does not have to be the universal set. The above example has white things as a superset of white birds, while the set containing both animate and inanimate objects is another possible universal set. A rectangle should be used to enclose the universal set, and other sets under discussion are enclosed inside. Relationships are indicated by overlapping regions.

Here, the English alphabet is our universal set. Vowels and consonants are nondisjoint subsets thereof. (Disjoint would mean their intersection was empty.)

The Greatest Mathematicians: Archimedes, Newton, Gauss

One of the goals of these lectures is to provide familiarity with the great mathematicians. We have already made reference to Whitehead, Russell, Gödel, Euclid, Pythagoras, Venn, and Euler above and will make reference to Bernoulli below. In this first lesson we will emphasize, however, the three greatest mathematicians of all time: Archimedes (c 287-212 BC), Newton (1642-1727), and Gauss (1777-1855) (c. is an abbreviation for the Latin word circa, meaning around.). Learning common Latin terms is another goal. Note that if there were a fourth greatest mathematician, it would be Euler.

Archimedes was born, lived, and died in Syracuse, Sicily but studied at Alexandria (Egypt)—at that time the center of learning. He is known as a mathematician, scientist, and inventor, but his greatest contributions were in geometry, such as the relationship between the surface area and volume of a sphere and its circumscribing cylinder. He found lower and upper limits for pi: by inscribing and circumscribing a circle with a regular 96-gon. He invented engines of war (mirrors, catapults, etc.) and the water screw. The principle of bouyancy named after him helped him determine whether or not a crown was pure gold—he streaked from the public bath shouting "Eureka, Eureka", or literally I found it, I found it. He is quoted as saying: "Give me a place to stand and I will move the earth"—meaning levers can do great feats. His methods of calculating areas in several cases were equivalent to calculus invented much later. Some of his works were lost and not all the stories and books attributed to him are necessarily accurate. One of your teachers has done extensive research an one such topic. Archimedes was drawing geometric figures in the sand when a Roman soldier, against specific orders, fatally struck him.

Sir Isaac Newton, tiny, weak, and not expected to survive his first day, was born in England on Christmas day (old style) 1642. He is known not only as one of the greatest mathematicians, but also one of the greatest physicists as well. He culminated (to climax) the scientific revolution and authored Principia, the most important single work in the history of modern science. Newton attended Trinity College, then laid the foundation of calculus and extended his ideas on color. He examined planetary motion and derived the inverse square law crucial to his theory of universal gravitation. The three laws of mechanics were named after him. He was also warden, then later master, of the mint. There he oversaw a great recoinage which included reeded edges on coins and tracking down a master counterfeiter. Two important quotes attributed to Newton are "If I have seen a little farther than others it is because I have stood on the shoulders of giants" and "I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me." Returning home from work at the Mint, Newton solved a mathematical problem that was given to European mathematicians to solve; he turned in his work the next day anonymously. Upon receiving the solution, Bernoulli exclaimed,"Ah! I recognize the lion by his paw." Newton was knighted for his scientific discoveries rather than deeds on the battlefield—a first. Newton was buried like a king in Westminster Abbey.

Carl Friedrich Gauss was German, born the only son of poor parents. However, his early genius was recognized as discussed in the next lesson at a young age. In his doctoral thesis at age 22, he developed the concept of complex numbers and the Fundamental Theorem of Algebra. He applied mathematics to gravitation, electricity, and magnetism, thus his name is closely tied into modern physics. Some of his important quotes are "Mathematics, the queen of the sciences, and arithmetic, the queen of mathematics" and "Pauca, sed matura (few, but ripe)." Gauss is perhaps most famous for what I like to rather redundantly call the bell-shaped, gaussian, normal curve which we will study later. He is also known for his method of least squares to obtain the best regression line which we will study much later.

 

 

 

List of Mathematicians

Several different mathematicians are referenced in this series of lectures and some have asked for a list. Here is a start.

 

 

 

List of Greek/Latin Terms

Several different Greek and Latin terms are purposefully used in this series of lectures and some have asked for a list. Here is a start.

 

 

 

 

 

 

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