Families of Sets and Extended Operations Families of Sets When dealing with sets whose elements are themselves sets it is fairly common practice to refer to them as families of sets, however this is not a definition. In fact, technically, a family of sets need not be a set, because we allow repeated elements, so a family is a multiset. However, we do require that when repeated elements appear they are distinguishable. F = {A , A , A , A } with A = {a,b,c}, A ={a}, A = {a,d} and 1 2 3 4 1 2 3 A = {a} is a family of sets. 4 Extended Union and Intersection Let F be a family of sets. Then we define: The union over F by: ∪ A={x :∃ A∈F x∈A}= {x :∃ A A∈F∧x∈A} A∈F and the intersection over F by: ∩ A = {x :∀ A∈F x∈A}= {x :∀ A A∈F ⇒ x∈A}. A∈F For example, with F = {A , A , A , A } where A = {a,b,c}, 1 2 3 4 1 A ={a}, A = {a,d} and A = {a} we have: 2 3 4 ∪ A = {a ,b , c , d } and ∩ A = {a}. A∈F A∈F Theorem 2.8 For each set B in a family F of sets, a) ∩ A ⊆ B A∈F b) B ⊆ ∪ A. A∈F Pf: a) Suppose x ∈ ∩ A, then ∀A ∈ F, x ∈ A. Since B ∈ F, we have x ∈ B. Thus, ∩ A ⊆ B. b) Now suppose y ∈ B. Since B ∈ F, y ∈ ∪ A. Thus, B ⊆ ∪ A. Caveat Care must be taken with the empty family F, i.e., the family containing no sets. Suppose the U is the universal set, and F is the empty family. Then by definition, ∩ A = {x∈U :∀ A A∈F ⇒ x∈A}= U . A∈F and ∪ A = {x∈U :∃ A A∈F∧x∈A}= ∅. A∈F Indexing Given a family of sets F, it is often convenient to associate to each set in the family a "label" called an index, which need not be related in any way to the elements of the set. The set of all indices, often denoted by ∆ is called an indexing set. Example: Consider the family F of half-open intervals of real numbers, [0,r). We could introduce indexing by defining A = [0,r) r and then referring to F by F = {A | r ∈ ℝ+}. ℝ+ is the indexing set r for this indexed family. Note: The indexing set may be finite or infinite, ordered or unordered (like the complex numbers). Although ℕ is often used for indexing infinite sets, not all infinite sets can be indexed by ℕ! DeMorgan's Laws = ∩ Aa ∪ Aa a∈ a∈ = ∪ Aa ∩ Aa a∈ a∈ Pf: x ∈ (∩A )c iff ~ (x ∈ ∩A ) iff a a ~( (∀ a) (a ∈ ∆ ⇒ x ∈ A )) iff (∃ a)(a ∈ ∆ ∧ x ∉ A ) iff a a c c (∃ a)(a ∈ ∆ ∧ x ∈ A ) iff x ∈ ∪ A . a a To prove the second law we can use the first. c cc c c cc c (∪ A ) = (∪ A ) = (∩ A ) = ∩ A a a a a Examples For each natural number n, let A = {1, 2, 3, ..., n}. n ∞ = ℕ ∪ Ai i=1 ∞ ∩ Ai = {1} i=1 5 A ∪ i = {1,2,3,4,5} i=1 7 A ∩ i = {1,2,3} i=3 Pairwise Disjoint Given a non-empty family of sets, there are two concepts of disjointness that can be applied (this does not arise if there are only two sets in the family). =∅ ∩ Aa a∈ An indexed family of sets F = {A } is pairwise disjoint iff for a a ∈ ∆ all a and b ∈ ∆, if A ≠ A , then A ∩ A = ∅. a b a b.