Mathematical Analysis I, 2004a

§§1–3. Sets and Operations on Sets. Quantifiers 3
The proof of (d) is sketched in Problem 1. The rest is left to the reader.
Because of (c), we may omit brackets in A ∪ B ∪ C and A ∩ B ∩ C; similarly
for four or more sets. More generally, we may consider whole families of sets,
i.e., collections of many (possibly infinitely many) sets. If M is such a family,
we define its union, ⋃ M, to be the set of all elements x, each belonging to at
least one set of the family. The intersection of M, denoted ⋂ M, consists of
those x that belong to all sets of the family simultaneously. Instead, we also
write
⋃
{X | X ∈M}and
⋂
{X | X ∈M}, respectively.
Often we can number the sets of a given family:
A 1 ,A 2 , ..., A n , ....
More generally, we may denote all sets of a family M by some letter (say, X)
with indices i attached to it (the indices may, but need not,benumbers).The
family M then is denoted by {X i } or {X i | i ∈ I}, wherei is a variable index
ranging over a suitable set I of indices (“index notation”). In this case, the
union and intersection of M are denoted by such symbols as
⋃
{Xi | i ∈ I} = ⋃ X i = ⋃ X i = ⋃ X i ;
i
i∈I
⋂
{Xi | i ∈ I} = ⋂ i
X i = ⋂ X i = ⋂ i∈I
X i .
If the indices are integers, wemaywrite
m⋃
X n ,
n=1
∞⋃
X n ,
n=1
m⋂
n=k
X n , etc.
Theorem 2 (De Morgan’s duality laws). For any sets S and A i (i ∈ I), the
following are true:
(i) S − ⋃ i
A i = ⋂ i
(S − A i ); (ii) S − ⋂ i
A i = ⋃ i
(S − A i ).
(If S is the entire space, we may write −A i for S − A i , − ⋃ A i for S − ⋃ A i ,
etc.)
Before proving these laws, we introduce some useful notation.
Logical Quantifiers. From logic we borrow the following abbreviations.
“(∀ x ∈ A) ...” means “For each member x of A, itistruethat....”
“(∃ x ∈ A) ...” means “There is at least one x in A such that ....”
“(∃! x ∈ A) ...” means “There is a unique x in A such that ....”