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 = {A1, A2, A3, A4} with A1 = {a,b,c}, A2 ={a}, A3 = {a,d} and
A4 = {a} is a family of sets.
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 = {A1, A2, A3, A4} where A1 = {a,b,c},
A2 ={a}, A3 = {a,d} and A4 = {a} we have:
∪ 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⊆B
b) B ⊆ ∪ A.
a)
A∈F
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 Ar = [0,r)
and then referring to F by F = {Ar | r ∈ ℝ }. ℝ is the indexing set
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
∩ ∪


A
=
∪ ∩A

Aa =
a∈
a∈
a
a∈
a∈

Aa
a
Pf: x ∈ (∩Aa)c iff ~ (x ∈ ∩Aa ) iff
~( (∀ a) (a ∈ ∆ ⇒ x ∈ Aa)) iff (∃ a)(a ∈ ∆ ∧ x ∉ Aa) iff
(∃ a)(a ∈ ∆ ∧ x ∈ Aac) iff x ∈ ∪ Aac.
To prove the second law we can use the first.
(∪ Aa)c = (∪ Aacc)c = (∩ Aac)cc = ∩ Aac
Examples
For each natural number n, let An = {1, 2, 3, ..., n}.
∞
∪A
∩A
∪A
∩A
i=1
∞
i=1
5
i=1
7
i=3
i
=ℕ
i
= {1}
i
= {1,2,3,4,5}
i
= {1,2,3}
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).
∩ A =∅
a∈
a
An indexed family of sets F = {Aa}a ∈ ∆ is pairwise disjoint iff for
all a and b ∈ ∆, if Aa ≠ Ab, then Aa ∩ Ab = ∅.