Basic Set Theory

BASIC SET THEORY James T. Smith San Francisco State University These notes outline some set theory on which many parts of mathematics are based. Sets The notions object, set, and membership are used in this theory without definition. The expression x 0 X indicates that the object x is a member of the set X. Any object with a member is a set, and sets are considered objects. Sometimes it’s assumed that sets are the only objects, but not in this outline. Notation Using gaudier letters for sets than for their members, as in g 0 G 0 G , often enhances clarity (but sometimes isn’t practical). These abbreviations are also useful: 0 ... ó ... = .... = / .... œ ... › ... is a member of is not a member of equals does not equal for all for some & ... w ... ¬ ... | ... ] ... and or not if ... then ... if and only if Equality Two sets are equal if and only if they have the same members: X = Y ] œt [ t 0 X ] t 0 Y ]. That is the extensionality principle. Extension Frequently, a set X is described by a statement of the form t0X ] Φ (*) 2008-01-29 16:02 Page 2 BASIC SET THEORY where Φ is a condition involving t. For example, in calculus you often consider intervals of real numbers: t 0 [a, b] ] a # t & t # b. If (*) holds, then X is called the extension of Φ. It’s appropriate to call X the extension, because, by the extensionality principle, you can deduce X = Y from (*) and the similar statement t 0 Y ] Φ. Since X is uniquely determined when (*) holds, the notation X = { t : Φ} is common. It’s read, “X is the set of all t such that Φ.” For the previous example, [a, b] = { t : a # t & t # b }. Each set X is the extension of some condition Φ —for example, the condition t 0 X. That is, t0X ] t0X X = { t : t 0 X }. However, there are conditions Φ that have no extension—that is, for which there’s no set X such that (*) holds. In 1902, Bertrand Russell discovered the most celebrated such condition: t ó t. If that had an extension X, then X 0 X ] X ó X, contradiction! One of the most important problems in foundations of mathematics is to determine which conditions have extensions. This outline, however, doesn’t attack that question. Frequently, it introduces new sets X as extensions of certain conditions. The assumptions that these particular conditions have extensions have never led to contradiction. Separation One type of condition always has an extension: a condition applying only to members of a previously given set. That is, to each set Y and each condition Φ corresponds a set X whose elements are those members t of Y that satisfy Φ: X = { t : t 0 Y & Φ}, abbreviated { t 0 Y : Φ}. This is the separation principle. It implies, for example, that the condition t = t has no extension: if there existed a set V such that t 0 V ] t = t, then Russell’s condition would have an extension, namely { t 0 V : t ó t}. This result can also be phrased, there’s no “universal” set that contains all objects. 2008-01-29 16:02 BASIC SET THEORY Page 3 Inclusion A set X is said to be included in a set Y —or called a subset of Y —if each member of X belongs also to Y: X f Y ] œt [ t 0 X | t 0 Y ]. This concept has the following properties: XfX XfY & YfX | X=Y XfY & YfZ | XfZ —reflexivity —weak antisymmetry —transitivity. Power Set To each set X corresponds a set the subsets of X: P X, called the power set of X, whose members are P X = { S : S f X }. Empty Set The condition t = / t has an extension, called the empty set φ: œt [t ó φ]. φ = {t : t = / t} By the extensionality principle, φ is the only set with no members. It’s a subset of every set. Singletons To each object x corresponds a set {x}, called singleton x, whose sole member is x: {x} = { t : t = x}. Pairs To any objects x and y corresponds a set {x, y}, called a pair, whose only members are x and y: {x, y} = { t : t = x w t = y}. 2008-01-29 16:02 Page 4 BASIC SET THEORY Notice that { x, y} = { y, x} { x, x} = { x}. Triples, quadruples, etc., could be introduced the same way, but more comprehensive methods will be presented later. Ordered Pairs To any two objects x and y corresponds an object <x, y>, called an ordered pair. For any objects x, y, xr, and yr, <x, y> = <xr, yr> ] x = xr & y = yr. Ordered triples and quadruples, etc., could be introduced the same way, but it’s easier to define <x, y, z> = <<x, y>, z> and extend that idea to quadruples, etc. Cartesian Product To any sets X and Y corresponds a set X × Y, called their Cartesian product, whose members are the ordered pairs whose first and second entries belong to X and Y: <x, y> 0 X × Y ] x 0 X & y 0 Y. Cartesian products of three or more sets are introduced as follows: X × Y × Z = (X × Y ) × Z <x, y, z> 0 X × Y × Z ] x 0 X & y 0 Y & z 0 Z. Relations A relation between two sets X and Y is a subset of X × Y. Thus φ and X × Y itself are relations between X and Y. This abbreviation is commonly used for relations R: x R y ] <x, y> 0 R . A relation between X and itself is called a relation on X. 2008-01-29 16:02 BASIC SET THEORY Page 5 Domain and Range To each relation R correspond two sets, called its domain and range, whose members are the first and second entries of the members of R: x 0 Dom R ] › y [x R y] y 0 Rng R ] › x [x R y]. Converse To each relation R between sets X and Y corresponds a relation R̆ between Y and X, called the converse of R, such that y R̆ x ] x R y. Relative Product If R is a relation between sets X and Y and S is a relation between sets Y and Z, then their relative product is the relation R*S defined as follows: x (R*S) z ] › y [x R y & y S z]. For example, if R is the relation of person to parent and S that of sibling to brother, then R*S is the relation of person to uncle. The following associative law is fundamental: for any R and S as described and any relation Q between sets W and X , (Q*R)*S = Q*(R*S). Proof. Suppose w ((Q*R)*S) z. Then ›y [w (Q*R) y & y S z ] b ›x [w Q x & x R y] ` b x (R*S) z ` b w (Q*(R*S)) z. Thus the left hand side of the associative law equation is included in the right hand side. You can demonstrate the reverse inclusion similarly. The associative law permits the abbreviation Q*R*S for (Q*R)*S or Q*(R*S). The following law is also important: for any relations R and S as described earlier, T = R*S | T̆ = S̆*R̆. 2008-01-29 16:02 Page 6 BASIC SET THEORY You can supply the proof. Identity To each set X corresponds the identity relation I X on X: x I X xr ] x = xr & x 0 X. For each relation R between X and a set Y, I X *R = R = R*IY . Image To each relation R and each set A corresponds a subset R[A] of Rng R called the image of A under R: y 0 R[ A] ] ›x [x 0 A & x R y]. For each relation S, (R*S)[ A] = S[R[ A]]. Functions A relation F is called a function from a set X to a set Y if Dom F = X, Rng F f Y and for all x, y, x F y & x F yr | y = yr. That is written F : X 6 Y. If F : X 6 Y and x 0 X then there is a unique y 0 Y such that x F y. This is written F : x ² y. That y is called the value of F at x, and it is denoted by F(x) or Fx . By the extensionality principle, two functions F and G are equal if and only if they have the same domain X and F(x) = G(x) for all x 0 X. The functions from a set X to a set Y constitute a set Y X. Here are some rules for manipulating these function sets: Y φ = { φ} X= / φ | φX = φ X f Y | IX 0 Y X. A function f : X 6 X is often called a singulary operation on X; a function g : X × X 6 X is often called a binary operation on X. A function is called constant if its range is a singleton. 2008-01-29 16:02 BASIC SET THEORY Page 7 Composition The composition of functions F : X 6 Y and G : Y 6 Z is the function G B F = F*G from X to Z. The associativity law holds: if also E : W 6 X, then (G B F ) B E = G B (F B E). That permits the abbreviation G B F B E for either of these compositions. The following manipulation rules hold: x 0 X | (G B F )(x) = G(F(x)) A f X | (G B F )[A] = G[F [A]] IY B F = F = F B I X . Injections If F : X 6 Y and F̆ is a function, then we say that F : X 6 Y injectively, and call F an injection. The following rules are helpful: φ : φ 6 Y injectively X f Y | IX : X 6 Y injectively F : X 6 Y injectively & G : Y 6 Z injectively | G B F : X 6 Z injectively. Surjections If F : X 6 Y and Rng F = Y, then we say that F : X 6 Y surjectively, and call F a surjection. The following rules hold: φ : φ 6 φ surjectively F : X 6 Y surjectively & G : Y 6 Z surjectively | G B F : X 6 Z surjectively. Bijections If F : X 6 Y injectively and surjectively, then we say that F : X 6 Y bijectively and call F a bijection. Here are useful rules: IX : X 6 X bijectively F : X 6 Y bijectively Y F̆ : Y 6 X bijectively F : X 6 Y bijectively & G : Y 6 Z bijectively Y G B F : X 6 Z bijectively. 2008-01-29 16:02 Page 8 BASIC SET THEORY A bijection from X to itself is called a permutation of X. The set X ! of all permutations of X is called the symmetric group on X. Inverse If F : X 6 Y bijectively, then F̆ is called the inverse of F and denoted by F –1. Here’s its most important property: F –1 B F = IX F B F –1 = IY . Proof. To show IX f F&1 B F, let x 0 X and define y = F(x). Then x F y, so that y F –1 x and hence x F *F –1 x, i.e. <x, x> 0 F –1 B F. To show F –1 B F f I X , let <x, xr> 0 F –1 B F, i.e. x F *F –1 xr. Then there exists y such that x F y and y F –1 xr. But x F y implies y F –1 x, and thus x = xr because F –1 is a function. The proof that F B F –1 = IY is similar. Here’s an important property of the inverse, complementary to the previous one: G : Y 6 X & G B F = IX | G = F –1 G : Y 6 X & F B G = IY | G = F –1 . Proof. Suppose G B F = I X ; then G = G B IY = G B (F B F –1 ) = (G B F ) B F –1 = I X B F –1 = F –1. The second result is proved similarly. Finally, if F : X 6 Y and G : Y 6 Z bijectively, then (G B F ) –1 = F –1 B G –1. Union To any sets X and Y corresponds a set X c Y called their union, whose members are the members of X and those of Y: t0XcY ] t0X w t 0 Y. The union is their “least upper bound” in the following sense: X, Y f X c Y X f Z & Y f Z | X c Y f Z. These commutative and associative laws hold: XcY=YcX X c (Y c Z) = (X c Y ) c Z. Associativity permits the abbreviation X c Y c Z for either side of this equation. 2008-01-29 16:02 BASIC SET THEORY Page 9 Intersection To any sets X and Y corresponds a set X 1 Y called their intersection, whose members are the elements common to X and Y: t 0 X 1 Y ] t 0 X & t 0 Y. The intersection is their “greatest lower bound” in the following sense: Z f X & Z f Y | Z f X 1 Y. X 1 Y f X, Y If X 1 Y = φ, then X and Y are called disjoint. These commutative, associative, and distributive laws hold: X1Y=Y1X X 1 (Y 1 Z) = (X 1 Y ) 1 Z X 1 (Y c Z) = (X 1 Y ) c (X 1 Z) X c (Y 1 Z) = (X c Y ) 1 (X c Z) . Associativity permits the abbreviation X 1 Y 1 Z for either side of the second equation. Union, Continued To any set X corresponds a set ^X called its union, whose members are the members of the members of X : t 0 ^X ] › X [ X 0 X & t 0 X ]. The union of X is its “least upper bound” in the following sense: œX [ X 0 X | X f ^X ] For any sets X and Y, ^{X, Y } = X c Y. œX [ X 0 X | X f Y ] | ^ X f Y. Intersection, Continued To any nonempty set X corresponds a set _ X called its intersection, whose members are the elements common to all members of X : t 0 _X ] œX [ X 0 X | t 0 X ]. The intersection of X is its “greatest lower bound” in the following sense: œX [ X 0 X | _X f X ] For any sets X and Y, _{X, Y } = X 1 Y. œX [ X 0 X | Y f X ] | Y f _X . 2008-01-29 16:02 Page 10 BASIC SET THEORY Union and Intersection, Continued Often you’ll be interested in the union or intersection of the range of a function X with domain I. This notation is common: ∪X i ∩X = { t : (›i 0 I)[ t 0 X i ]} i±I i = { t : (œi 0 I)[ t 0 X i ]}. i±I When the set I is clear from the context, these are usually abbreviated as ^ i X i and _ i X i . This notation simplifies the statements of many rules—for example, the distributive laws A 1 ^ i X i = ^ i (A 1 X i ) A c _ i X i = _ i (A c X i ). Relative Complement In this paragraph, all sets are assumed to be subsets of a single set U. To each such set X corresponds a set –X, its complement (relative to U ), whose members are those elements of U not in X: –X = {t 0 U : t ó X }. These rules hold: –U = φ – –X = X X c –X = U X f Y ] –Y f –X – ∪X i i±I = ∩ ( X i) i± I –φ = U —double negation X 1 –X = φ —contraposition – ∩X i±I i = ∪ ( X i) —de Morgan i± I Natural Numbers There’s a set whose members are called natural numbers. Among its members is φ, which in this context is called zero and written 0. There’s a bijection S: 6 {n 0 :n= / 0} called the successor operation, which satisfies the first principle of recursive proof: 00Xf & œn [ n 0 X | S(n) 0 X ] | X = . From these considerations follows— by a complicated argument—the first principle of recursive definition: 2008-01-29 16:02 BASIC SET THEORY Page 11 given any set Y, any y 0 Y, and any function G : Y 6 Y, there’s a unique function F : 6 Y such that | F (S(n)) = G(F (n)) ]. F (0) = y & œn [ n 0 Binary sum and product operations + and @ and an order relation # on are defined, and their usual properties proved, following standard recursive methods. In particular, 1 is defined as S(0), so that S(n) = n + 1 for all n 0 ; and 2 is defined as 1 + 1. A second principle of recursive proof is sometimes handier than the first: every nonempty X f x 0 X. contains a member w such that w # x for all There’s a corresponding second principle of recursive definition: given any set Y and any function G : × P ( × Y ) 6 Y, there’s a unique F : 6 Y such that for each n 0 , œn [ n 0 | F (n) = G(<n, {<m, F (m)> : m 0 & m < n}>) ]. Integers Following standard algebraic procedures, integers are defined as certain sets of ordered pairs of natural numbers, and the familiar arithmetic operations are constructed for them. They form an ordered integral domain in which each nonempty set of nonnegative elements has a minimum element. All such domains are isomorphic. Rational Numbers Again following standard algebraic procedures, rational numbers are defined as certain sets of ordered pairs of integers, and the familiar arithmetic operations are constructed for them. They form a prime ordered field . All such fields are isomorphic. Real Numbers Following standard analytic procedures, real numbers are defined as certain sets of sequences of rational numbers—i.e. certain sets of functions from to —and the familiar arithmetic operations are constructed for them. They form a complete ordered field . All such fields are isomorphic. (Alternative definitions of real numbers as certain sets of rational numbers or certain sequences of integers are common.) 2008-01-29 16:02 Page 12 BASIC SET THEORY Complex Numbers Following a standard algebraic procedure, complex numbers are defined as pairs of real numbers, and the familiar arithmetic operations are constructed for them. They form an algebraically closed field . Trivial questions 1. Pφ=? P {φ} = ? 2. {x, x} = ? 3. φ×X=? X×φ=? 4. Dom φ = ? Rng φ = ? 5. Dom {<x, y>} = ? Rng {<x, y>} = ? 6. Dom ( X × Y ) = ? Rng (X × Y ) = ? 7. φ̆ = ? R = {<x, y>} | R̆ = ? 8. R = X × Y | R̆ = ? R = S̆ | R̆ = ? 9. φ*R = ? R*φ = ? 10. {<x, y>}*{<y, z>} = ? 11. (X × Y )*(Y × Z) = ? Careful! 12. Dom(R*S) f Dom(?) Rng(R*S) f Rng(?) 13. Iφ = ? I{x} = ? 14. φ[A] = ? R[φ] = ? 15. R[Dom R] = ? R̆[Rng R] = ? 16. Xcφ=? X1φ=? 17. XcX=? X1X=? 18. {x} c { y} = ? {x} 1 { y} = ? 19. XcY=Y ] ? X1Y=Y ] ? 20. X c (X 1 Y ) = ? X 1 (X c Y ) = ? 21. IX [A] = ? 22. ^φ = ? 23. ^{X } = ? 24. Ac ∪X i±I i _{X } = ? = ∪ (?) i±I A1 ∩X i±I i = ∩ (?) i±I 2008-01-29 16:02 BASIC SET THEORY Page 13 Routine exercises 1. Prove œS [ x 0 S | y 0 S ] | x = y. 2. Prove X f Y | P X f P Y. 3. Prove {{x}, {x, y}} = {{xr}, {xr, yr}} | x = xr & y = yr (Kuratowski, 1921). The notion of ordered pair can be defined this way. 4. Prove X × Y = Y × X ] φ = X 5. Suppose X f Xr. What can you say about the relationship of a. b. c. d. e. f. 6. X=Y w Y = φ. X × Y and Xr × Y, Y × X and Y × Xr ? R[X] and R[Xr] ? X c Y and Xr c Y, Y c X and Y c Xr ? X 1 Y and Xr 1 Y, Y 1 X and Y 1 Xr ? ^ X and ^ Xr ? _ X and _ Xr ? Suppose R and Rr are relations and R f Rr. What can you say about the relationship of a. b. c. d. 7. w Dom R and Dom Rr, Rng R and Rng Rr ? R̆ and R̆r ? R*S and Rr*S, S*R and S*Rr ? R[A] and Rr[A] ? Suppose œi [ X i f Yi ]. What can you say about the relationship of a. b. ^ i X i and ^ i Yi ? _ i Xi and _ i Yi ? 8. Prove that if R and S are relations, and Q = R*S, then Q̆ = S̆ *R̆. Prove that if F : X 6 Y and G : Y 6 Z bijectively, then (G B F ) –1 = F –1 B G –1. 9. Prove that the composition of two injections is an injection. Do the same for surjections and bijections. 10. Prove that if R and S are relations, then (R*S)[A] = S[R[A]]. Prove that if F : X 6 Y and G : Y 6 Z then (G B F )[A] = G[F [A]] . 11. Prove that if R is a relation, then (X × Y )*R = X × R[Y ]. What’s R*(X × Y )? 12. Prove all the distributive laws mentioned. 2008-01-29 16:02 Page 14 BASIC SET THEORY 13. Prove the de Morgan laws. 14. Prove the modular law X f Z ] X c (Y 1 Z) = (X c Y ) 1 Z. Prove that X c Y = X c Z & X 1 Y = X 1 Z | Y = Z. 15. a. b. c. d. 16. When does a relative complement of a set equal that set itself ? 17. a. b. Why can’t we define _ φ? Why can’t we define an absolute complement –X = {t : t ó X } ? 18. a. Prove that if R is a relation, then R[ ^ i A i ] = ^ i R[A i ]. Prove ^ P X = X. Prove X f P ^ X . Find X so that X = P ^ X . / P ^X . Find X so that X = Suppose S is a relation and for each i, Ri is a relation. Prove b. c. d. e. 19. Dom ^ i Ri = ^ i Dom Ri and Rng ^ i Ri = ^ i Rng Ri R = ^ i Ri | R̆ = ^ i R̆i (^ i Ri)*S = ^ i (Ri*S) and S*^ i Ri = ^ i (S*Ri ) (^ i Ri )[A] = ^ i Ri [A] . If F : X 6 Y, then define F̆“ : P Y 6 P X by setting F̆“(B) = F̆ [B] for every B 0 P Y. Prove that if F is injective, then F̆“ is surjective. Prove that if F is surjective, then F̆“ is injective. Substantial problems 1. Undertake routine exercise 18 with unions replaced by intersections. You’ll find that you must replace many equations by inclusions. In those cases, find examples where the equations hold, and examples where they don’t. Keep the examples simple—use intersections of two sets only. 2. Let R be a relation. Prove that B f Rng R | B f R[R̆[B]] . Prove that R is a function if and only if œB [ B f Rng R | B = R[R̆[B]] ]. Find a function F and a set B f Dom F such that B = / F̆ [F [B]], and another F and B such that the equation does hold. 2008-01-29 16:02 BASIC SET THEORY 3. Page 15 A function F : X 6 Y is called right cancellative if G : Y 6 Z & Gr : Y 6 Zr & G B F = Gr B F | G = Gr. It’s left cancellative if E : W 6 X & Er : Wr 6 X & F B E = F B Er | E = Er. Prove that F is injective if and only if it’s left cancellative and surjective if and only if it’s right cancellative. 4. Suppose that for each i 0 I, Fi is a function from a subset of a set X to a set Y. Further, assume (œi, j 0 I)(›k 0 I)[ Fi f Fk & Fj f Fk]. Prove that ^ i Fi is a function from a subset of X to Y, and if each Fi is injective, then so is ^ i Fi . 5. Consider some sets A n for n 0 Liminf n A n = ∪∩ A n . Define Limsup n A n = m n–m ∩∪ A n . m n–m Prove a. b. c. 6. Liminf n A n f Limsup n A n œn [ A n f A n+1 ] | Liminf n A n = ^ n A n = Limsup n A n œn [ A n+1 f A n ] | Liminf n A n = _ n A n = Limsup n A n . Let m, n 0 and X and Y be sets with m and n elements. How many elements have the sets (X × {0}) c (Y × {1}) X×Y PX Y X? How many injections are there from X to Y? How many bijections are there from X to X? 7. Show that this condition on t has no extension: ¬›s [ s 0 t & t 0 s ]. References Bourbaki, Nicholas. 1968. Theory of Sets. Elements of Mathematics, Part 1, Book 1. Reading, Massachusetts: Addison–Wesley. LC: QA248.B73413. Pages 65–277. Skip over all references to his (rather unusual) axiom system. Cohen, Leon W., and Gertrude Ehrlich. 1963. The Structure of the Real Number System. Princeton: D. Van Nostrand. LC: QA241.C67. Lipschutz, Seymour. 1964. Schaum’s Outline of Theory and Problems of Set Theory and Related Topics. New York: McGraw–Hill. LC: QA248.L57. ISBN: 0-070-38159-3. Pages 1–184. Upper-division text. 2008-01-29 16:02 Page 16 BASIC SET THEORY Monk, J. Donald. 1969. Introduction to Set Theory. New York: McGraw–Hill. LC: QA248.M53. Axiomatic treatment, von Neumann–Bernays–Gödel–Morse version. Beginning graduate level. Stoll, Robert R. [1963] 1979. Set Theory and Logic. New York: Dover Publications. Originally published by W. H. Freeman. LC: QA248.S7985. ISBN: 0-486-63829-4. Pages 1–154, 289–306. Between the levels of Lipschutz and Monk. Suppes, Patrick. 1960. Axiomatic Set Theory. Princeton: D. Van Nostrand. LC: QA248.S92. Axiomatic treatment, Zermelo–Fraenkel version. Same level as Monk 1969. Rather close to this outline. 2008-01-29 16:02