Topology 01: Set Theory and Logic Lingerois Shanghai Jiao Tong University Update: January 10, 2020 1 Preliminarys of Set Theory We skip naive set theory, one could refer to the book for a review. We introduce a term for set theory. We say set A and B intersects if A \ B , ;. Note Vacuously True: If p is false, for any q, p ) q is always true. We say this kind of proposition is vacuously true. One can comprehend the truth of p ) q by using the notion of set theory. Let A = fxjp¹xº = 1g, and B = fxjq¹xº = 1g, then p¹xº ) q¹xº is saying that if x 2 A then x 2 B, which is equal to A ⊂ B. For x < A (equal to p¹xº = 0), of course x < B (equal to q¹xº = 0). Proposition 1.1 p ) q is equal to :q ) p. Proof. The only case that p ) q is false is when p is true and q is false, which implies :q ):p is false. A similar argument can prove :q ):p is true then p ) q is true. Theorem 1.2 DeMorgan’s Law A − ¹B [ Cº = ¹A − Bº [ ¹A − Cº A − ¹B \ Cº = ¹A − Bº \ ¹A − Cº The set of sets is called collection, denoted by A; B; ··· . The collection of all the subsets of A is called the power set, denoted by P¹Aº. We define the the union and intersection of all sets in A. For given A, the union of elements in A is defined as Ø A = fxj9A 2 A; x 2 Ag A2A And the intersection of A’s elements is defined as Ù A = fxjxg A2A A problem here is what if A = ;? One can prove that the union is still well defined if A =. But for the intersection, "8A 2 A; x 2 A" is vacuously true for any X. Therefore we don’t define the intersection of empty A. We skip the discuss of naive Cartesian product. 2 Function We define the function f : A ! B as a subset of C × D. We define it like this to circumvent the vague description of "corresponding". Definition 2.1 A rule of assignment is a subset r of C × D, where for any ¹a; bº;¹a; b0º 2 r, we have b = b0. The definition is saying that r can only contain at most one element ¹a; bº for any a 2 C. We say C is the domain of r; 2 D is the codomain of r; fbj9¹a; bº 2 rg ⊂ D is the range of r, which is also called the image set of r. Remark We don’t use the term "range" for B of f : A ! B, otherwise, we use "codomain" to keep in line with other notes. Definition 2.2 A function f is a rule of assignment r, together with a domain B that contains the image set of r. The domain of r is called the domain of f , the image set of r is called the image set of f . B is called the codomain of f . Definition 2.3 For functions f : A ! B and g : B ! C, the composition of f and g, denoted by f ◦ g, is a function f ◦ g : A ! C, defined as ¹ f ◦ gº¹aº = f ¹g¹aºº. We skip the definition of injection, surjection and bijection, by shown them by fig.1. (a) injection (b) surjection (c) bijection (d) not an injection, nor a surjection Figure 1: Compare of Functions Remark We use "injective", "surjective" and "bijective" for adjectives. We never use ambiguous words like "one-to-one" and "onto". Some notions are used frequently. For function f : X ! Y, let A ⊂ X and B ⊂ Y, we define The image of A as f ¹Aº = f f ¹xºjx 2 Ag; The preimage of B as f −1¹Bº = fxj f ¹xº 2 Bg. 3 By defining them, we have Proposition 2.4 We have the following holds: 1. x 2 f −1¹Bº () f ¹xº 2 B; 2. f −1¹ f ¹Aºº ⊂ A; 3. f ¹ f −1º¹Bº ⊃ B; 4. f ¹A \ Bº ⊂ f ¹Aº \ f ¹Bº. We illustrate the possible non-equality of 2. and 3. by fig.2. (a) f −1¹f ¹Aºº ( A (b) f ¹f −1º¹Bº ) B Figure 2: Compare of Functions 3 Relation Definition 3.1 A relation on a set A is a subset C of A × A. 3.1 Equivalence Relation We skip the definition of equivalence relation and partition. Note When we defines equivalence class and partition, we never says the equivalence classes C of some relation C on A is an partition of A. This fact is true, but one has to prove it. To prove this, one has to show that (1) every x 2 A lies in some C 2 C; (2) any x 2 A never lies in two distinct classes C1;C2 2 C. 4 3.2 Order Definition 3.2 An order relation (or simple order, linear order) C of set A is a relation that satiesfy the follows: 1. For any x; y 2 A such that x , y, we have x ∼ y or y ∼ x; 2. There exists no x 2 A such that x ∼ x; 3. If x ∼ y and y ∼ z, then x ∼ z. The order relation of a set is saying that one can line up the elements of A by a thread, in the order of the order relation. We often use "<" instead of "∼" to denote the relations in an order relation. Note There’s no parlance of "strict simple order" or "non-strict simple order". A simple order never includes x ∼ x for any x 2 A. The notion x ≤ y means one of the following two statements holds: x and y are different elements in A and x < y; x and y are different notions of the same elements in A. Definition 3.3 For a set X, let < be the simple order on X. For a < b, we denote fxja < x < bg as ¹a; bº, and we say it is a open interval. If ¹a; bº is empty, we say b is the immediate successor of a. Definition 3.4 For set A and B has simple order <A and <B correspondingly. We say A; B has the same order type if there exists an bijection f : A ! B such that a1 <A a2 ) f ¹a1º <B f ¹a2º As for A; B with simple order, we can define a simple order for A × B. Definition 3.5 For set A and B has simple order <A and <B correspondingly. Define the simple order < on A × B: when a1 <A a2 or when a1 = a2 and b1 <B b2, ¹a1; b1º < ¹a2; b2º We say it is the dictionary order relation. 5 3.3 Least Upper Bound Property We now discuss least upper bound property, which is a amazing as well as useful property used frequently in analysis. Let’s define something first. Definition 3.6 Let A be a set with simple order <. Let A0 be a subset of A. We say b is the largest element in A0 if b0 2 A0 and for any x 2 A0 it holds that x ≤ b0. Similarly, we say b is the smallest element in A0 if for any x 2 A0 it holds that b ≤ x. Remark In fact the names with "largest" and "smallest" are quite straightforward since we use "maximum" and "minimal" more often. Definition 3.7 We say A0 ⊂ A is bounded above if there exists b 2 A such that for all x 2 A0 it holds that x ≤ b; We say b is a upper bound of A0. If there’s a smallest element in the collection of the upper bounds of A, we say it is the least upper bound (or supremum) of A0, denoted by sup A0. Similarly, one could define bounded below, lower bound, greatest lower bound. Definition 3.8 We say a set A with simple order has least upper bound property if for all its nonempty subset A0, if A0 has an upper bound, it has least upper bound. Similarly, we cound define greatesst lower bound property. Notice that discussing all (from the start of section 3.3) these concepts is significant only if the order is specified. The simple order can be omitted if there’s no ambiguity. An amazing of the previous definition is that the least upper bound property and the greatest lower bound property is exactly the same property. Theorem 3.9 Ordered set A has least upper bound property if and only if it has greatest lower bound property. We postpone the proof of theorem 3.9 in the notes of Mathematical Analysis. 6 4 Integers and Real Numbers We skip the definition of binary operations. Definition 4.1 A set R we called real numbers, is a field (with addition + and multiplication ·), with a simple order < and the following properties (assuming x; y 2 R): 1. If x > y, then x + z > y + z; 2. If x > y and z > 0, then x · z > y · z; 3. Simple order > has least upper bound property; 4. If x < y, there exists z 2 R such that x < z < y. We say a field is an ordered field if it satisfies properties 1. and 2.. We say a set with simple order is an linear continuum if it satisfies properties 3. and 4.. Note we don’t prove such R exists, since it is too complex.