A Brief History of Sets

8/9/2019 A Brief History of Sets

1/6

A brief history of setsA set is an unordered collection of objects, and as such a set isdetermined by the objects it contains. Before the 19th century itwas uncommon to think of sets as completed objects in their own

right. Mathematicians were familiar with properties such as being anatural number, or being irrational, but it was rare to think of saythe collection of rational numbers as itself an object. (There wereexceptions. From Euclid mathematicians were used to thinking ofgeometric objects such as lines and planes and spheres which wemight today identify with their sets of points.)In the mid 19th century there was a renaissance in Logic. Forthousands

of years, since the time of Aristotle and before, learned individualshad beenfamiliar with syllogisms as patterns of legitimate reasoning, forexample:All men are mortal. Socrates is a man. Therefore Socrates ismortal.But syllogisms involved descriptions of properties. The idea of

pioneers such as Boole was to assign a meaning as a set to these

descriptions. For example, the two descriptions is a man and isa male homo sapiens both describe the same set, viz. the set of allmen. It was this objectification of meaning, understanding

properties as sets, that led to a rebirth of Logic and Mathematics inthe 19th century. Cantor took the idea of set to a revolutionarylevel, unveiling its true power. By inventing a notion of size of sethe was able compare different forms of infinity and, almostincidentally, to shortcut several traditional mathematicalarguments.

But the power of sets came at a price; it came with dangerousparadoxes.The work of Boole and others suggested a programme exposited

by Frege, and Russell and Whitehead, to build a foundation for allof Mathematics on Logic.


2/6

Though to be more accurate, they were really reinventing Logic inthe process,and regarding it as intimately bound up with a theoryof sets. The paradoxes of set theory were a real threat to thesecurity of the foundations. But with a lot of worry and care the

paradoxes were sidestepped, first by Russell and Whiteheadstheory of stratified types and then more elegantly, in for examplethe influential work of Zermelo and Fraenkel. The notion of set isnow a cornerstone of Mathematics.The precise notion of proof present in the work of Russell andWhitehead laid the scene for Godels astounding result of 1931:any sound proof system able to deal with arithmetic willnecessarily be incomplete, in the sense that it will be impossible to

prove all the statements within the system which are true. Godelstheorem relied on the mechanical nature of proof in order to beable to encode proofs back into the proof system itself. After aflurry of activity, through the work of Godel himself, Church,Turing and others, it was realised by the mid 1930s that Godelsincompleteness result rested on a fundamental notion, that ofcomputability. Arguably this marks the birth of Computer Science.Motivation Why learn Set Theory? Set Theory is an importantlanguage and tool for reasoning. Its a basis for Mathematics

pretty much all Mathematics can be formalised in Set Theory. Whyis Set Theory important for Computer Science?Its a useful tool for formalising and reasoning about computationand the objects of computation. Set Theory is indivisible fromLogic where Computer Science has its roots. It has been and islikely to continue to be a a source of fundamental ideas inComputer Science from theory to practice; Computer Science,

being a science of the artificial, has had many of its constructs and

ideas inspired by Set Theory. The strong tradition, universality andneutrality of Set Theory make it firm common ground on which to

provide unification between seemingly disparate areas andnotations of Computer Science. Set Theory is likely to be aroundlong after most present-day programming languages have fadedfrom memory. A knowledge of Set Theory should facilitate your


3/6

ability to think abstractly. It will provide you with a foundation onwhich to build a firm understanding and analysis of the new ideasin Computer Science that you will meet.

SetsSets. A set is a collection of objects, called elements of theset. A set can be represented by listing its elements between

braces:A = {1, 2, 3, 4, 5}. The symbol 2 is used to express that an elementis

(Or belongs to) a set, for instance 3 2 A. Its negation is representedby62, e.g. 7 62 A. If the set is finite, its number of elements isrepresented|A|, e.g. if A = {1, 2, 3, 4, 5} then |A| = 5.Some important sets are the following:1. N = {0, 1, 2, 3, } = the set of natural numbers.12. Z = { , 3, 2, 1, 0, 1, 2, 3, } = the set of integers.3. Q = the set of rational numbers.4. R = the set of real numbers.5. C = the set of complex numbers.Is S is one of those sets then we also use the following notations:21. S+ = set of positive elements in S, for instanceZ+ = {1, 2, 3, } = the set of positive integers.2. S = set of negative elements in S, for instanceZ = {1,2,3, } = the set of negative integers.3. S_ = set of elements in S excluding zero, for instance R_ = the

set of non zero real numbers.Set-builder notation. An alternative way to define a set, calledsetbuilder

Notation, is by stating a property (predicate) P(x) verified byExactly its elements, for instance A = {x 2 Z | 1 _ x _ 5} = set of


4/6

1Note that N includes zerofor some authors N = {1, 2, 3, },without zero.2When working with strings we will use a similar notation with adifferent

meaningbe careful not to confuse it.

integers x such that 1 _ x _ 5i.e.: A = {1, 2, 3, 4, 5}. In general:A = {x 2 U | p(x)}, where U is the universe of discourse in whichthe

predicate P(x) must be interpreted, or A = {x | P(x)} if the universeof discourse for P(x) is implicitly understood. In set theory thetermuniversal set is often used in place of universe of discourse for agiven

predicate.3Principle of Extension. Two sets are equal if and only if they havethe same elements, i.e.:A = B _ 8x (x 2 A $ x 2 B) .Subset. We say that A is a subset of set B, or A is contained inB, and we represent it A _ B, if all elements of A are in B, e.g., ifA = {a, b, c} and B = {a, b, c, d, e} then A _ B.

A is a proper subset of B, represented A _ B, if A _ B butA 6= B, i.e., there is some element in B which is not in A.Empty Set. A set with no elements is called empty set (or null set,or void set), and is represented by ; or {}.

Note that nothing prevents a set from possibly being an element ofanother set (which is not the same as being a subset!). For instanceif A = {1, a, {3, t}, {1, 2, 3}} and B = {3, t}, then obviously B isan

element of A, i.e., B 2 A.Power Set. The collection of all subsets of a set A is called the

power set of A, and is represented P(A). For instance, if A = {1, 2,3},thenP(A) = {;, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3},A} .


5/6

Exercise: Prove by induction that if |A| = n then |P(A)| = 2n.Multisets. Two ordinary sets are identical if they have the sameelements, so for instance, {a, a, b} and {a, b} are the same set

because

they have exactly the same elements, namely a and b. However, insome applications it might be useful to allow repeated elements inaset. In that case we use multisets, which are mathematical entitiessimilar to sets, but with possibly repeated elements. So, asmultisets,{a, a, b} and {a, b} would be considered different, since in the firstone

the element a occurs twice and in the second one it occurs onlyonce.3Properly speaking, the universe of discourse of set theory is thecollection ofall sets (which is not a set).

Properties of Sets.The set operations verify the followingproperties:1. Associative Laws:A [ (B U C) = (A U B)U[ CA \ (B C) = (A B) C2. Commutative Laws:A U B = BU AA B = B A3. Distributive Laws:A U (B C) = (A U B) (A U C)

A (B U C) = (A B) U (A C)4. Identity Laws:A U = AA U = A5. Complement Laws:A U A = UA A =;6. Idempotent Laws:


6/6

A UA = AA A = A7. Absorption Laws:AU (A B) = AA (A U B) = A8. Involution Law:A = A

A Brief History of Sets

Documents

Transcript of A Brief History of Sets