Chapter 6 Set Theory

1. Chapter 6 Set Theory

2. 6.1 Basic Definitions Introduction: Roughly speaking, a set is a collection of objects that satisfy a certain property, but in set theory, the words “set” and “element” are intentionally left as undefined. There is also another undefined relation , called the “membership” relation. If S is a set and a is an element of S, then we write aS, and we can say that abelongs to S.

3. 6.1 Basic Definitions Introduction: The { } notation: If a set M has only a finite number of elements say, 3, 7, and 11, then we can write M = {3, 7, 11} (the order in which they appear is unimportant.) A set can also be specified by a defining property, for instance S = {x  : -2 < x < 5} (this is almost always the way to define an infinite set).

4. 6.1 Basic Definitions Introduction: Some basic axioms Axiom of pairing – For any a and b, there exists a set {a, b} that contains exactly a and b. Axiom of extension – For any two sets A and B, In other words, two sets are equal if and only if they have the same elements. For example, {1, 2, 3} = {3, 1, 2} = {1, 2, 3, 2}

5. Axioms of Zermelo-Fraenkel • Axiom of Extensionality • Axiom of Pairing • Axiom Schema of Separation.If  is a property with parameter p, then for any set X, there exists a set Y = { u  X: (u, p)} • Axiom of UnionFor any set X, there is a set Y that is the union of all elements in X. • Axiom of Power SetFor any set X, there exists a set Y that contains all subsets of X. • Axiom of InfinityThere is an infinite set. • Axiom Schema of ReplacementIf F is a function, then for any set X, there is a set Y = {F(x): xX} • Axiom of RegularityEvery non-empty set has an -minimal element. • Axiom of ChoiceFor every non-empty set S of non-empty sets, there is at least one choice function for S.

6. Choice Function Let S be a collection of non-empty sets, then a choice function for S is a function such that f(X) is an element in X. S f X1 X2 X4 X3 X5

7. A very bizarre consequence of the Axiom of choice. The Banach-Tarski paradox (1924) It is possible to dissect the solid 3D unit sphere into 6 pieces such that these pieces can be reassembled by rigid motions (rotations and translations only) to form two solid unit spheres. Of course these pieces are non-measurable (i.e. has no volume and boundary in the usual sense) and are very complicated. In fact, only 5 pieces will be enough (Robinson 1947) and this is minimal. One of those pieces is a single point that will be used as the center of one of the two new spheres.

8. 6.1 Basic Definitions Subsets: Given two sets A and B, we say that A is a subset of B, denoted by if In other words, A is a subset of B if all elements in A are also in B. Example: (see next page)

9. An example of subset: Let E be the English alphabet, hence it is a set of 26 letters E = {a, b, c, … , x, y , z} In the Hawaiian alphabet however, it contains only 7 consonants H = { a, e, i, o, u, h, k, l, m, n, p, w} Hence H is a subset of E.

10. 6.1 Basic Definitions ( ) ( ) = Û Í Ù Í A B A B B A Ì A B Í ¹ A B and A B Lemma: Two sets A and B are equal if A is a subset of B and B is a subset of A. i.e. Proper Subsets: Given two sets A and B, we say that A is a proper subset of B, denoted by if In other words, A is a proper subset of B if all elements in A are also in B but A is “smaller” than B.

11. Exercises Determine whether each statement is true or false. True a) 3 {1,2,3} f) {2}  {1, {2},{3}} False b) 1  {1} False g) {1}  {1, 2} True c) {2}  {1, 2} False h) 1  {{1}, 2} False True i) {1}  {1, {2}} True d) {3}  {1, {2}, {3}} e) 1  {1} True j) {1}  {1} True

12. 6.1 Basic Definitions Operations of Sets: Let A and B be two subsets of a larger set U, we can define the following, 1. Union of A and B, 2. Intersection of A and B, 3. Difference of B minus A, 4. Complement of A,

13. 6.1 Basic Definitions Ordered pairs: For any two elements a and b (not necessarily distinct), we define (a, b) = {a, {a, b}} which is called the ordered pair of a and b. It is not hard to see that (a, b) = (c, d) if and only if (a = c and b = d) In other words, the order of the elements is important. We can similarly define ordered triples and ordered n-tuples (x1, x2, x3, … , xn)

14. 6.1 Basic Definitions Cartesian Products: For any two sets A and B, the Cartesian Product of A and B, denoted by A×B (read A cross B), is the set of all ordered pairs of the form (a, b) where aA and bB. Given sets A1, A2 , … , An we can define the Cartesian product A1 × A2 × · · ·× An as the set of all ordered n-tuples.

15. Definition of whole numbers The number 0 is defined as the empty set Ø. The number 1 is defined as the set {Ø}. The number 2 is defined as the set {Ø}  {{Ø}} = {Ø, {Ø}} … … In general, the natural number n + 1 is defined as the set n{n} We can prove by induction that the set representing n has exactly n elements. Remark: Every natural number n (as a set) is transitive, i.e. every element in n is also a subset of n. In other words, n contains all the elements of its elements, etc.

16. Symmetric Difference The symmetric difference of two sets A and B is the set A  B = (A – B )  (B – A) = (A  B) – (A∩B) A B Two set A and B are almost equal to each other if A  B is a small set, such as being measure 0 or meager.

17. 6.1 The empty set, Partition, and Power sets Definition: A set with no element in it is called an empty set. Theorem If Ø is an empty set, then it is a subset of any set. Corollary There is only one empty set. Notation Since there is only one empty set, we use the symbol Ø to denote this unique empty set, and we will call it the empty set.

18. Do you know? The equivalency of the empty set in music is a rest, and a rest is undeniably a musical note by definition, but it is hard to believe that someone will go to the extreme and create a piece of music using only rests. This is the controversial musical piece called 4'33" - by the American experimental composer John Cage (1912 – 1992)

19. The premiere of the three-movement 4'33" was given by David Tudor on August 29, 1952, at Woodstock, New York as part of a recital of contemporary piano music. The audience saw him sit at the piano, and lift the lid of the piano. Some time later, without having played any notes, he closed the lid. A while after that, again having played nothing, he lifted the lid. And after a period of time, he closed the lid once more and rose from the piano. The piece had passed without a note being played, in fact without Tudor or anyone else on stage having made any deliberate sound, although he timed the lengths on a stopwatch while turning the pages of the score. It is not known however, that any one requested an encore.

20. Interesting Facts: The 4’33” has been recorded on several occasions. An ‘orchestral version’ given by the BBC Symphony Orchestra was broadcast on BBC Radio 3 in January 2004. A story tells that a 7” vinyl version of 4’33” was at one time popular on the juke boxes of a number of bars, as it gave customers a relief from an otherwise relentless soundtrack of rock and roll. Today, you can download a free mp3 version of this piece from many websites.

21. Actually Cage’s creation was influenced by other works in the visual arts. The "white paintings' produced by Rauschenberg at Black Mountain in 1951 Ad Reinhardt “Abstract painting” 1963, 60”× 60” oil on canvas.

22. 6.1 The empty set, Partition, and Power sets Definition: Two sets A and B are said to be disjoint if their intersection is empty. Proposition: Given any two sets A and B, Definition: A collection of sets A1, A2 , … An , are said to be mutually disjoint (or pairwisely disjoint) if

23. Partitions of Sets Definition: A collection of non-empty sets {A1, A2 , …, An} is a partition of a set A if

24. Power sets and their properties Definition: Given a set A, the power set of A, denoted , is the set of all subsets of A. (the existence of such a power set is an axiom when A is infinite, and cannot be proved from other previous axioms) Theorem: For all sets A and B, Theorem: If A is a set with n elements, then has 2n elements.

25. 6.2 Properties of Sets Theorem 5.2.1: 1. Inclusion of Intersection; For all sets A and B, 2. Inclusion in Union: For all sets A and B, 3. Transitive property of Subsets: For all sets A, B, and C,

26. 6.2 Properties of Sets Theorem 5.2.2: Set Identities Let all sets referred to below be subsets of a set U. 1. Commutative Laws: For all sets A and B 2. Associative Laws: For all sets A, B, and C, 3. Distributive Laws: For all sets A, B, and C,

27. 6.2 Properties of Sets Ç = A U A C C = ( A ) A Ç = È = A A A and A A A Theorem 5.2.2: Set Identities Let all sets referred to below be subsets of a set U. 4. Intersection with U: For all sets A 5. Double complement Law: For all sets A 6. Idempotent Laws: For all sets A,

28. 6.2 Properties of Sets C C C C C C È = Ç Ç = È ( A B ) A B and (A B) A B È = A U U È Ç = Ç È = A ( A B ) A and A ( A B ) A C - = Ç A B A B Theorem 5.2.2: Set Identities Let all sets referred to below be subsets of a set U. 7. DeMorgan’s Law: For all sets A and B, 8. Union with U: For all sets A 9. Absorption Laws: For all sets A and B, 10. Alternate Representation for set difference: For all sets A and B,

29. Properties of the Empty Set Let all sets referred to below be subsets of a set U. 1. Union with Ø: 2. Intersection and union with the complement 3. Intersection with Ø 4. Complement of U and Ø