Why Read This Book? - Index of

3.3 Families of Sets 75
x ∈ �
A ⇔ (∃A ∈ F)(x ∈ A) (3.24)
A∈F
The form of (3.23) says there exists a course number α ∈ A such that graduate x
is on the roster of mathematics course number α. The form of (3.24) says simply
that there is a mathematics course roster on which the name of graduate x appears,
without any reference to a course number. With this, we arrive at the following
definition.
Definition 3.3.3 Let F be a family of sets indexed by A. Then the union over
F is defined by
�
Aα ={x : (∃α ∈ A)(x ∈ Aα)} (3.25)
α∈A
Without reference to the index set, this becomes
�
A ={x : (∃A ∈ F )(x ∈ A)} (3.26)
A∈F
Now what does ∩α∈AAα (or ∩A∈F A) correspond to in our metaphor? This is
the intersection of all rosters, so it is the list of all graduates who passed all mathematics
courses at PU. So what does it mean mathematically to say x ∈∩α∈AAα (or
x ∈∩FA)? x ∈ �
Aα ⇔ (∀α ∈ A)(x ∈ Aα) (3.27)
α∈A
x ∈ �
A ⇔ (∀A ∈ F)(x ∈ A) (3.28)
A∈F
With this we arrive at the following definition.
Definition 3.3.4 Let F be a family of sets indexed by A. Then the intersection
over F is defined by
�
Aα ={x : (∀α ∈ A)(x ∈ Aα)} (3.29)
α∈A
Without reference to the index set, this becomes
�
A ={x : (∀A ∈ F )(x ∈ A)} (3.30)
A∈F
Many of the results we proved in Section 3.2 for a family of two or three sets
carry over to analogous results for a family of sets of any size. So as you work