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Naïve Set Theory Basic Definitions Naïve Set Theory Is the Non-Axiomatic Treatment of Set Theory

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Naïve Set Theory Basic Definitions Naïve Set Theory Is the Non-Axiomatic Treatment of Set Theory Naïve Set Theory Basic Definitions Naïve set theory is the non-axiomatic treatment of set theory. In the axiomatic treatment, which we will only allude to at times, a set is an undefined term. For us however, a set will be thought of as a collection of some (possibly none) objects. These objects are called the members (or elements) of the set. We use the symbol "∈" to indicate membership in a set. Thus, if A is a set and x is one of its members, we write x ∈ A and say "x is an element of A" or "x is in A" or "x is a member of A". Note that "∈" is not the same as the Greek letter "ε" epsilon. Basic Definitions Sets can be described notationally in many ways, but always using the set brackets "{" and "}". If possible, one can just list the elements of the set: A = {1,3, oranges, lions, an old wad of gum} or give an indication of the elements: ℕ = {1,2,3, ... } ℤ = {..., -2,-1,0,1,2, ...} or (most frequently in mathematics) using set-builder notation: S = {x ∈ ℝ | 1 < x ≤ 7 } or {x ∈ ℝ : 1 < x ≤ 7 } which is read as "S is the set of real numbers x, such that x is greater than 1 and less than or equal to 7". In some areas of mathematics sets may be denoted by special notations. For instance, in analysis S would be written (1,7]. Basic Definitions Note that sets do not contain repeated elements. An element is either in or not in a set, never "in the set 5 times" for instance. When repeated elements are needed, we refer to multisets, but they are not sets. Definition: Let A and B be sets. We say that A is a subset of B iff every element of A is also an element of B. Symbolically, A ⊆ B ⇔ (∀x)( x∈A ⇒ x∈B ). Observe that "being a subset" is a proposition ... a statement that is either true or false. Ex: A = {0,2,5} If B = ℤ then A ⊆ B but if B = ℕ, A ⊈ B. Basic Definitions Notice the following distinctions: Let A = {0,2, Elsie the Cow} Is A a subset of B if B is: B = {0,1,2,3, lions, Elsie the Cow} Yes B = {0,1,2, lions, cows} No B = {0,2, a lion, a cow} Maybe Definition: Two sets, A and B are equal iff every element of one is also an element of the other. Symbolically, A = B ⇔ (∀x)( x∈A ⇔ x∈B ). More useful in proofs than this definition is the logically equivalent statement: Basic Definitions (∀x)( x∈A ⇔ x∈B ) ≡ (∀x)( x∈A ⇒ x∈B ) ∧ (∀x)( x∈A ⇐ x∈B ) Which can be written in set notation (as opposed to logic notation) as: A = B iff A ⊆ B and B ⊆ A. Definition: For sets A and B, we use the notation A ⊂ B, to mean that A is a subset of B and A ≠ B. Note that this is not a universally accepted notation. Some authors have used this symbol to mean subset as we have defined it. The Empty Set In the axiomatic development of set theory there is one (and only one) existence axiom and it is: Axiom: The empty set exists. Definition: Let ∅ = {x | x ≠ x }. This set contains no elements and is called an empty set. Notice that any contradiction can be used in the defining clause. Also, the symbol "∅" is not the Greek letter "φ" phi, it is actually of Danish origin. To emphasize the definition we have given, notice that the predicate x ∈ ∅ is always false. The Empty Set Thm: The empty set is unique. Pf: Suppose that A and B are empty sets. Since A is an empty set, the statement x∈A is false for all x, so (∀x)( x∈A ⇒ x∈B ) is true! That is, A ⊆ B. Since B is an empty set, the statement x∈B is false for all x, so (∀x)( x∈Β ⇒ x∈Α ) is also true. Thus, B ⊆ Α. Since A ⊆ B and B ⊆ Α we have A = B. We have used a direct proof here ... but had we decided to do this by contradiction we would have .... The Empty Set Thm: The empty set is unique. Pf: BWOC suppose that A and B are distinct empty sets. Since A is an empty set, the statement x∈A is false for all x, so (∀x)( x∈A ⇒ x∈B ) is true! That is, A ⊆ B. Since B is an empty set, the statement x∈B is false for all x, so (∀x)( x∈Β ⇒ x∈Α ) is also true. Thus, B ⊆ Α. Since A ⊆ B and B ⊆ Α we have A = B. →← The contradiction method in this case does not add anything to the argument, it just puts an unnecessary layer over the basic direct method ... this is not wrong, its just in bad taste!! The Universal Subset Thm: The empty set is a subset of every set.* Pf: Let A be a set. Since x ∈ ∅ is false for all x, (∀x)( x ∈ ∅ ⇒ x∈Α ) is true. Thus, ∅ ⊆ A. Since A was arbitrary, ∅ is a subset of every set. ∗ Observe that the statement is a quantified statement: (∀A, A a set)( ∅ ⊆ A) To prove a "for all" statement, one starts with an arbitrary element of universe for the variable ... assuming no properties other than membership, and show that the statement is true for it. Warnings Some minor points concerning the empty set: 1. ∅ is the empty set, {∅} is NOT the empty set. {∅} is a set that contains 1 element (i.e., the empty set), if you want to use set brackets to describe the empty set, use "{}". 2. While ∅ is a subset of every set, it is rarely ever a member of a set (to be so, it must be in the list of elements). Thus, ∅ ∈ {1, Elsie, ∅} as well as ∅ ⊆ {1, Elsie, ∅} but ∅ ∉ {1, Elsie, {∅}} Text Problems T T T T F T F F F F T T Theorem 2.1(b) Thm: For every set A, A ⊆ A. Pf: Let A be an arbitrary set. Since x∈A ⇒ x∈A for all x (whether x∈A is true or not) (∀x)(x∈A ⇒ x∈A ) is true. So, A ⊆ A for all sets A. Theorem 2.2 Thm: Let A, B and C be sets. If A ⊆ B and B ⊆ C, then A ⊆ C. Pf: Let x ∈ A. Since A ⊆ Β, x ∈ B. Since B ⊆ C, x ∈ C. Thus, for all x, x ∈ A ⇒ x ∈ C. So, A ⊆ C. We refer to this result as the "transitivity of subset inclusion". Proper Subsets Definition: For any set A, ∅ and A are called improper subsets of A. All other subsets are called proper subsets. Example: Let S = {1, 2, Elsie} The proper subsets of S are: {1}, {2}, {Elsie} {1, 2}, {1, Elsie}, {2, Elsie} The improper subsets of S are: ∅, {1,2, Elsie} Power Sets Definition: Given a set S, the power set of S, denoted P (S), is the set consisting of all the subsets of S. P (S) = {B | B ⊆ S}. Important! The elements of the power set of a set are themselves sets, both proper and improper. If S = {0,2,5} then {5} ∈ P (S) but 5 ∉ P (S). On the other hand, 5 ∈ S and {5} ∉ S. Example: If S = {1, 2, Elsie} then P (S) = {∅, {1}, {2}, {Elsie},{1, 2}, {1, Elsie}, {2, Elsie}, S} Power Sets Thm: If a set S has n elements then P (S) has 2n elements. We will provide at least two proofs of this important result in later sections when we talk about the techniques used. Example: If S = ∅ then P (∅) = {∅} which is not the empty set! S contains 0 elements, and P (∅) has 20 = 1 element. Power Sets Thm: Let A and B be sets. Then A ⊆ B iff P (A) ⊆ P (B). Pf: (⇒ Sufficiency) Let C ∈ P (A), then C ⊆ A. Since A ⊆ Β we have by Thm 2.2 that C ⊆ B. Thus, C ∈ (B). Since C was arbitrary, P (A) ⊆ P (B). (⇐ Necessity) Let x ∈ A. Then {x} ⊆ A. Since P (A) ⊆ P (B), {x} ∈ P (B), so {x} ⊆ B. Thus x ∈ B, and since x was arbitrary, A ⊆ B. Both parts of this proof illustrate the method of "element chasing" used frequently in proofs involving sets. Power Sets Thm: Let A and B be sets. Then A ⊆ B iff P (A) ⊆ P (B). An alternate proof the the necessity of this theorem can be given which does not involve element chasing: (⇐ Necessity) Since A ∈ P (A) and P (A) ⊆ P (B), we have that A ∈ P (B). Thus, A ⊆ B. Compare this with the previous proof: (⇐ Necessity) Let x ∈ A. Then {x} ⊆ A. Since P (A) ⊆ P (B), {x} ∈ P (B), so {x} ⊆ B. Thus x ∈ B, and since x was arbitrary, A ⊆ B. Text Problems T T T T T T T F T T T F P ({∅, {∅}}) = { ∅, {∅, {∅}}, {∅}, {{∅}} }. Set Operations Definitions: Let A and B be sets. The union of A and B, denoted A ∪ B, is the set A ∪ B = {x | x ∈ A ∨ x ∈ B}. The intersection of A and B, denoted A ∩ B, is the set A ∩ B = {x | x ∈ A ∧ x ∈ B}. The relative complement of B in A (difference) denoted by A – B, is the set A – B = {x | x ∈ A ∧ x ∉ B}. Venn Diagrams Frequently, one sees diagrams used to represent the relationships between sets, called Venn diagrams. For instance: Venn diagrams should be used with care, and you should never base an argument on what such a picture looks like, since one diagram can not represent all possible relationships.
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