Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Indexed families of sets
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of March 10, 2017

2. A (singly) indexed family of sets over T is any function (association), s.t. with
any member of t ∈ T we associate a set Xt; it is denoted {Xt}t∈T .
For instance, {Xn}n∈N, where Xn = {k ∈ N : k < n}, is an indexed family of sets.
We have: X0 = ∅, X1 = {0}, X2 = {0, 1}, X3 = {0, 1, 2}, ...
For instance, {Xa}a∈R+ , where R+ = {x ∈ R : x > 0} and Xa = (0, 1 + a), is an
indexed family of open intervals of reals. We have, X1
2
= (0, 11
2), etc.
If T = {t ∈ N : t < n} we write {Xt}t<n .
If T = {t ∈ Z : k t n} we write {Xt}n
t=k .
If T = {t ∈ Z : k t} we write {Xt}∞
t=k .

3. The union of {Xt}t∈T , denoted t∈T Xt , is {Xt : t ∈ T}
– the union of the corresponding family of sets.
Let T =∅. The intersection of {Xt}t∈T , denoted t∈T Xt , is {Xt : t ∈ T}
– the intersection of the corresponding family of sets.
If T = {t ∈ Z : 0 t < n} we write t<n Xt and t<n Xt (if n > 0).
If T = {t ∈ Z : k t n} we write n
t=k Xt and n
t=k Xt (if n k).
If T = {t ∈ Z : k t} we write ∞
t=k Xt and ∞
t=k Xt .
Fact
1. x ∈ t∈T Xt iff ∃t∈T x ∈ Xt
2. x ∈ t∈T Xt iff ∀t∈T x ∈ Xt

4. Example
Let Xn = (−1
n , 1 + 1
n ), for n ∈ Z+.
This is an indexed family of open intervals of real numbers.
t∈Z+ Xn = ?

5. Example
Let Xn = (−1
n , 1 + 1
n ), for n ∈ Z+.
This is an indexed family of open intervals of real numbers.
X1 =(−1, 2), X2 =(−1
2, 11
2), X3 =(−1
3, 11
3), ..., X10 =(− 1
10, 1 1
10), ...
t∈Z+ Xn = ?

6. Example
Let Xn = (−1
n , 1 + 1
n ), for n ∈ Z+.
This is an indexed family of open intervals of real numbers.
X1 =(−1, 2), X2 =(−1
2, 11
2), X3 =(−1
3, 11
3), ..., X10 =(− 1
10, 1 1
10), ...
(−1, 2) ⊃ (−1
2, 11
2) ⊃ (−1
3, 11
3) ⊃ ... ⊃ (− 1
10, 1 1
10) ⊃ ...
t∈Z+ Xn = ?

7. Example
Let Xn = (−1
n , 1 + 1
n ), for n ∈ Z+.
This is an indexed family of open intervals of real numbers.
X1 =(−1, 2), X2 =(−1
2, 11
2), X3 =(−1
3, 11
3), ..., X10 =(− 1
10, 1 1
10), ...
(−1, 2) ⊃ (−1
2, 11
2) ⊃ (−1
3, 11
3) ⊃ ... ⊃ (− 1
10, 1 1
10) ⊃ ...
t∈Z+ Xn = (−1, 2)

8. Example
Let Xn = (−1
n , 1 + 1
n ), for n ∈ Z+.
This is an indexed family of open intervals of real numbers.
X1 =(−1, 2), X2 =(−1
2, 11
2), X3 =(−1
3, 11
3), ..., X10 =(− 1
10, 1 1
10), ...
(−1, 2) ⊃ (−1
2, 11
2) ⊃ (−1
3, 11
3) ⊃ ... ⊃ (− 1
10, 1 1
10) ⊃ ...
t∈Z+ Xn = (−1, 2)
t∈Z+ Xn = ?

9. Example
Let Xn = (−1
n , 1 + 1
n ), for n ∈ Z+.
This is an indexed family of open intervals of real numbers.
X1 =(−1, 2), X2 =(−1
2, 11
2), X3 =(−1
3, 11
3), ..., X10 =(− 1
10, 1 1
10), ...
(−1, 2) ⊃ (−1
2, 11
2) ⊃ (−1
3, 11
3) ⊃ ... ⊃ (− 1
10, 1 1
10) ⊃ ...
t∈Z+ Xn = (−1, 2)
0, 1 ∈ (−1, 2), 0, 1 ∈ (−1
2, 11
2), 0, 1 ∈ (−1
3, 11
3), ..., 0, 1 ∈ (− 1
10, 1 1
10), ...
t∈Z+ Xn = ?

10. Example
Let Xn = (−1
n , 1 + 1
n ), for n ∈ Z+.
This is an indexed family of open intervals of real numbers.
X1 =(−1, 2), X2 =(−1
2, 11
2), X3 =(−1
3, 11
3), ..., X10 =(− 1
10, 1 1
10), ...
(−1, 2) ⊃ (−1
2, 11
2) ⊃ (−1
3, 11
3) ⊃ ... ⊃ (− 1
10, 1 1
10) ⊃ ...
t∈Z+ Xn = (−1, 2)
0, 1 ∈ (−1, 2), 0, 1 ∈ (−1
2, 11
2), 0, 1 ∈ (−1
3, 11
3), ..., 0, 1 ∈ (− 1
10, 1 1
10), ...
t∈Z+ Xn = [0, 1]

11. Unions and intersections of indexed subfamilies
Let T0 ⊆ T. Then, {Xt}t∈T0 is an indexed subfamily of {Xt}t∈T .
Fact
Let T0 ⊆ T. Then:
1. t∈T0
Xt ⊆ t∈T Xt
2. t∈T0
Xt ⊇ t∈T Xt
Exercise
1. Disprove: if T0 ⊂ T then t∈T0
Xt ⊂ t∈T Xt
2. Disprove: if T0 ⊂ T then t∈T0
Xt ⊃ t∈T Xt

12. Indexed families vs. families
Let X be a family of sets.
The indexed family of sets corresponding to X is {x}x∈X .
Let {Xt}t∈T be an indexed family of sets.
The family of sets corresponding to {Xt}t∈T is {Xt : t ∈ T}.
Unlike with (plain) families, in an indexed family an element may have repetitions:
{Xn}n∈{1,2,3}, where X1 = {1}, X2 = ∅, X3 = {1}.
The corresponding family of sets is {∅, {1}}, and it has only two members.
Fact
1. Let {Xt}t∈T be an indexed family of sets, and X the corresponding family of sets.
Then, t∈T Xt = X and t∈T Xt = X, if T =∅.
2. Let X be a family of sets, and {Xt}t∈T the corresponding indexed family of sets.
Then, t∈T Xt = X and t∈T Xt = X, if T =∅.
So, families of sets and indexed families of sets can be treated interchangingly.

13. Analogies
is a finite disjunction : i∈{1,2,3} Ai ⇔ A1 ∨ A2 ∨ A3, etc.
is a finite conjunction : i∈{1,2,3} Ai ⇔ A1 ∧ A2 ∧ A3, etc.
In set theory, is a generalization of ∪, and is a generalization of ∩.
In logic, is a generalization of ∨, and is a generalization of ∧.
We think informally/intuitively/semantically:
∃x∈X A(x) is a (possibly infinite) disjunction x∈X A(x), and
∀x∈X A(x) is a (possibly infinite) conjunction x∈X A(x).
We will explore analogies among , ∃ , and among , ∀ , .
We will look at particular cases involving ∨, ∪ and ∧, ∩.

14. Proposition
Ai ⊆ B, for every i ∈ I iff i∈I Ai ⊆ B.
Ai ⊇ B, for every i ∈ I iff i∈I Ai ⊇ B (assuming I =∅).
Analogous statements
∀v (A(v) → B) ⇔ (∃v A(v)) → B (assuming v ∈ var(B)).
∀v (A(v) ← B) ⇔ (∀v A(v)) ← B (assuming v ∈ var(B)).
i∈I (Ai → B) ⇔ ( i∈I Ai ) → B.
i∈I (Ai ← B) ⇔ ( i∈I Ai ) ← B.
A particular case of the item above, with I = {1, 2}:
(A1 → B) ∧ (A2 → B) ⇔ (A1 ∨ A2) → B.
(A1 ← B) ∧ (A2 ← B) ⇔ (A1 ∧ A2) ← B.
A particular case of the Proposition above, with I = {1, 2}:
A1 ⊆ B and A2 ⊆ B iff A1 ∪ A2 ⊆ B.
A1 ⊇ B and A2 ⊇ B iff A1 ∩ A2 ⊇ B.

15. Proposition
Ai0 ⊆ i∈I Ai , for any index i0 ∈ I.
Ai0 ⊇ i∈I Ai , for any index i0 ∈ I (assuming I =∅).
Analogous statements
A[ t
v ] ⇒ ∃v A , for any term t.
A[ t
v ] ⇐ ∀v A , for any term t.
Holds because the universe is assumed to be non-empty.
Ai0 ⇒ i∈I Ai , for any index i0 ∈ I.
Ai0 ⇐ i∈I Ai , for any index i0 ∈ I.
A particular case of the item above with I = {1, 2}:
A1 ⇒ A1 ∨ A2 and A2 ⇒ A1 ∨ A2
A1 ⇐ A1 ∧ A2 and A2 ⇐ A1 ∨ A2
A particular case of the Proposition above, with I = {1, 2}:
A1 ⊆ A1 ∪ A2 and A2 ⊆ A1 ∪ A2
A1 ⊇ A1 ∩ A2 and A2 ⊇ A1 ∩ A2

16. Proposition
i∈I Ai ⊆ i∈I Ai (assuming I =∅).
Analogous statements
∀v A ⇒ ∃v A.
It holds because the universe of quantification is always assumed to be non-empty.
i∈I Ai ⇒ i∈I Ai
A particular case of the item above with I = {1, 2}:
A1 ∧ A2 ⇒ A1 ∨ A2
A particular case of the Proposition above, with I = {1, 2}:
A1 ∩ A2 ⊆ A1 ∪ A2

17. Proposition [De Morgan laws]
Let {Ai }i∈I be a non-empty indexed family of subsets of U. Then:
( i∈I Ai )c
= i∈I Ai
c
.
( i∈I Ai )c
= i∈I Ai
c
.
Analogous statements
¬∀v∈X A ⇔ ∃v∈X ¬A.
¬∃v∈X A ⇔ ∀v∈X ¬A.
They hold because the universe of quantification is assumed to be non-empty.
¬ i∈I Ai ⇔ i∈I ¬Ai .
¬ i∈I Ai ⇔ i∈I ¬Ai .
A particular case of the item above with I = {1, 2}:
¬(A1 ∧ A2) ⇔ ¬A1 ∨ ¬A2.
¬(A1 ∨ A2) ⇔ ¬A1 ∧ ¬A2.
A particular case of the Proposition above, with I = {1, 2}:
(A1 ∩ A2)c
= A1
c
∪ A2
c
.
(A1 ∪ A2)c
= A1
c
∩ A2
c
.

18. Proposition [distributivity laws 1]
1. i∈I (A ∪ Bi ) = A ∪ ( i∈I Bi ).
i∈I (A ∩ Bi ) = A ∩ ( i∈I Bi ) (assuming I =∅).
2. i∈I (A ∩ Bi ) = A ∩ ( i∈I Bi ).
i∈I (A ∪ Bi ) = A ∪ ( i∈I Bi ) (assuming I =∅).
Statements analogous to item 2.
∃v (A ∧ B(v)) ⇔ A ∧ ∃v B(v) (assuming v ∈ var(A)).
∀v (A ∨ B(v)) ⇔ A ∨ ∀v B(v) (assuming v ∈ var(A)).
i∈I (A ∧ Bi ) ⇔ A ∧ ( i∈I Bi ).
i∈I (A ∨ Bi ) ⇔ A ∨ ( i∈I Bi ).
A particular case of the item above with I = {1, 2}:
(A ∧ B1) ∨ (A ∧ B2) ⇔ A ∧ (B1 ∨ B2).
(A ∨ B1) ∧ (A ∨ B2) ⇔ A ∨ (B1 ∧ B2).
A particular case of the Proposition item 2, with I = {1, 2}:
(A ∩ B1) ∪ (A ∩ B2) ⇔ A ∩ (B1 ∪ B2).
(A ∪ B1) ∩ (A ∪ B2) ⇔ A ∪ (B1 ∩ B2).
Exercise. Write statements analogous to the Proposition item 1.

19. Proposition [distributivity laws 2]
1. i∈I (Ai ∪ Bi ) = ( i∈I Ai ) ∪ ( i∈I Bi ).
i∈I (Ai ∩ Bi ) = ( i∈I Ai ) ∩ ( i∈I Bi ) (assuming I =∅)
2. i∈I (Ai ∩ Bi ) ⊆ ( i∈I Ai ) ∩ ( i∈I Bi ).
i∈I (Ai ∪ Bi ) ⊇ ( i∈I Ai ) ∪ ( i∈I Bi ) (assuming I =∅)
Statements analogous to item 2.
∃v (A ∧ B) ⇒ ∃v A ∧ ∃v B.
∀v (A ∨ B) ⇐ ∀v A ∨ ∀v B.
i∈I (Ai ∧ Bi ) ⇒ ( i∈I Ai ) ∧ ( i∈I Bi ).
i∈I (Ai ∨ Bi ) ⇐ ( i∈I Ai ) ∨ ( i∈I Bi ).
A particular case of the item above with I = {1, 2}:
(A1 ∧ B1) ∨ (A2 ∧ B2) ⇒ (A1 ∨ A2) ∧ (B1 ∨ B2).
(A1 ∨ B1) ∧ (A2 ∨ B2) ⇐ (A1 ∧ A2) ∨ (B1 ∧ B2).
A particular case of the Proposition item 2, with I = {1, 2}:
(A1 ∩ B1) ∪ (A2 ∩ B2) ⊆ (A1 ∪ A2) ∩ (B1 ∪ B2).
(A1 ∪ B1) ∩ (A2 ∪ B2) ⊇ (A1 ∩ A2) ∪ (B1 ∩ B2).
Exercise. Write statements analogous to the Proposition item 1.

20. The remaining slides present an extra credit material.

21. A doubly indexed family of sets over S and T is any function (association),
s.t. with any ordered pair s, t ∈ S × T we associate a set Xs,t;
it is denoted {Xs,t}s∈S,t∈T .
If the two index sets are the same, {Xs,t}s∈T,t∈T is denoted {Xs,t}s,t∈T and is called
doubly indexed family over T .
1. Xm,n = {k ∈ N : k < mn}, for m ∈ {1, 2} and n ∈ {1, 2, 3},
is a doubly indexed family of sets. We have:
n = 1 n = 2 n = 3
m = 1 X1,1 = {0} X1,2 = {0, 1} X1,3 = {0, 1, 2}
m = 2 X2,1 = {0, 1} X2,2 = {0, 1, 2, 3} X2,3 = {0, 1, 2, 3, 4, 5}
2. Xp,n = {kp : k ∈ N and k > n}, for p ∈ Primes and n ∈ N,
is a doubly indexed family of sets.
We have X3,4 = {15, 18, 21, 24, 27, 30, ...}, etc.

22. Every doubly indexed family of sets over S and T can be viewed as
a singly indexed family over T of singly indexed families over S:
{Xs,t}s∈S,t∈T versus {{Xs,t}s∈S }t∈T .
Let us call {Xs,t}s∈S (for any fixed t) a component family of {{Xs,t}s∈S }t∈T .
The union of a doubly indexed family {Xs,t}s∈S,t∈T on the first index
is the singly indexed family over T of the unions of such component families.
Similarly with the intersections.
The union of {Xs,t}s∈S,t∈T on the first index , denoted s∈S Xs,t ,
is the singly indexed family of sets { s∈S Xs,t}t∈T .
Let S =∅. The intersection of {Xs,t}s∈S,t∈T on the first index , s∈S Xs,t ,
is the singly indexed family of sets { s∈S Xs,t}t∈T .

23. Also, every doubly indexed family of sets over S and T can be viewed as
a singly indexed family over S of singly indexed families over T:
{Xs,t}s∈S,t∈T versus {{Xs,t}t∈T }s∈S .
Let us call {Xs,t}t∈T (for any fixed s) a component family of {{Xs,t}t∈T }s∈S .
The union of a doubly indexed family {Xs,t}s∈S,t∈T on the second index
is the singly indexed family over S of the unions of such component families.
Similarly with the intersections.
The union of {Xs,t}s∈S,t∈T on the second index , denoted t∈T Xs,t ,
is the singly indexed family of sets { t∈T Xs,t}s∈S .
Let S =∅. The intersection of {Xs,t}s∈S,t∈T on the second index , t∈T Xs,t ,
is the singly indexed family of sets { t∈T Xs,t}s∈S .

24. Example
Let S = {1, 2} and T = {7, 8, 9}.
Let {Xs,t}s∈S,t∈T be a doubly indexed family of sets over S and T.
t = 7 t = 8 t = 9
s = 1 X1,7 X1,8 X1,9
s = 2 X2,7 X2,8 X2,9
Let: T7 = X1,7 ∪ X2,7, T8 = X1,8 ∪ X2,8, T9 = X1,9 ∪ X2,9.
Let: S1 = X1,7 ∪ X1,8 ∪ X1,9, S2 = X2,7 ∪ X2,8 ∪ X2,9.
t = 7 t = 8 t = 9
s = 1 X1,7 X1,8 X1,9 S1
s = 2 X2,7 X2,8 X2,9 S2
T7 T8 T9
s∈S Xs,t = { s∈S Xs,t}t∈T = {Tt}t∈T .
t∈T Xs,t = { t∈T Xs,t}s∈S = {Ss}s∈S .

25. Example, continued
S = {1, 2} and T = {7, 8, 9}.
T7 = X1,7 ∪ X2,7, T8 = X1,8 ∪ X2,8, T9 = X1,9 ∪ X2,9.
S1 = X1,7 ∪ X1,8 ∪ X1,9, S2 = X2,7 ∪ X2,8 ∪ X2,9.
t = 7 t = 8 t = 9
s = 1 X1,7 X1,8 X1,9 S1
s = 2 X2,7 X2,8 X2,9 S2
T7 T8 T9
Notice that:
x ∈ t∈T s∈S Xs,t iff
x ∈ t∈T Tt iff
x ∈ T7 ∩ T8 ∩ T9 iff
x ∈ (X1,7∪, X2,7) ∩ (X1,8∪, X2,8) ∩ (X1,9∪, X2,9) iff
∀t∈T ∃s∈S x ∈ Xs,t.

26. Fact
1. x ∈ s∈S t∈T Xs,t iff ∃s∈S ∃t∈T x ∈ Xs,t
2. x ∈ t∈T s∈S Xs,t iff ∃t∈T ∃s∈S x ∈ Xs,t
3. x ∈ s∈S t∈T Xs,t iff ∀s∈S ∀t∈T x ∈ Xs,t (assuming S, T =∅)
4. x ∈ t∈T s∈S Xs,t iff ∀t∈T ∀s∈S x ∈ Xs,t (assuming S, T =∅)
5. x ∈ s∈S t∈T Xs,t iff ∃s∈S ∀t∈T x ∈ Xs,t (assuming T =∅)
6. x ∈ t∈T s∈S Xs,t iff ∃t∈T ∀s∈S x ∈ Xs,t (assuming S =∅)
7. x ∈ s∈S t∈T Xs,t iff ∀s∈S ∃t∈T x ∈ Xs,t (assuming S =∅)
8. x ∈ t∈T s∈S Xs,t iff ∀t∈T ∃s∈S x ∈ Xs,t (assuming T =∅)
Corollary
s∈S t∈T Xs,t = t∈T s∈S Xs,t.
s∈S t∈T Xs,t = t∈T s∈S Xs,t (assuming S, T =∅).
Instead of s∈T t∈T Xs,t we will write s,t∈T Xs,t .
Instead of s∈T t∈T Xs,t we will write s,t∈T Xs,t (assuming S, T =∅).

27. Proposition
Let {Ai,j }i∈I,j∈J be a non-empty doubly indexed family over I and J.
The following diagram shows all the inclusions among sets obtained by applying two
big union or intersection operations.
i∈I j∈J Ai,j
1 
//
j∈J i∈I Ai,j! u
''
i∈I,j∈J Ai,j
AÙ
77
1 
//
! u
''
i∈I,j∈J Ai,j
j∈J i∈I Ai,j
1 
//
i∈I j∈J Ai,j
AÙ
77

28. Proposition
Let {Ai,j }i,j∈I be a non-empty doubly indexed family over I.
This means, {Ai,j }i∈I,j∈I , the two index sets being the same.
The following diagram shows all the inclusions among sets obtained by applying two
big union or intersection operations.
i∈I j∈I Ai,j
1  //
! u
((
j∈I i∈I Ai,jt
''
i,j∈I Ai,j
B
77
1  //
t
''
i∈I Ai,i
AÙ
66
1  //
! u
((
i∈I Ai,i
1  //
i,j∈I Ai,j
j∈I i∈I Ai,j
AÙ
66
1  //
i∈I j∈I Ai,j
B
77

29. An analogy with quantifiers
i∈I j∈I Ai,j
1  //
! u
((
j∈I i∈I Ai,jt
''
i,j∈I Ai,j
B
77
1  //
t
''
i∈I Ai,i
AÙ
66
1  //
! u
((
i∈I Ai,i
1  //
i,j∈I Ai,j
j∈I i∈I Ai,j
AÙ
66
1  //
i∈I j∈I Ai,j
B
77
∃v ∀w A(v, w) +3
$,
∀w ∃v A(v, w)
$,
∀v,w A(v, w)
2:
+3
$,
∀v A(v, v)
2:
+3
$,
∃v A(v, v) +3 ∃v,w A(v, w)
∃w ∀v A(v, w)
2:
+3 ∀v ∃w A(v, w)
2:

30. Analogies explored
Conisder the inclusion and the implication on the top of the two diagrams.
i∈I j∈J Ai,j ⊆ j∈J i∈I Ai,j
∃v ∀w A(v, w) ⇒ ∀w ∃v A(v, w)
i∈I j∈J Ai,j ⇒ j∈J i∈I Ai,j
Here is a particular case of the item above with I = {1, 2} and J = {7, 8, 9}.
(A1,7 ∧A1,8 ∧A1,9)∨(A2,7 ∧A2,8 ∧A2,9) ⇒ (A1,7 ∨A2,7)∧(A1,8 ∨A2,8)∧(A1,9 ∨A2,9)
which can be easier to read when written as:
(A1 ∧ B1 ∧ C1) ∨ (A2 ∧ B2 ∧ C2) ⇒ (A1 ∨ A2) ∧ (B1 ∨ B2) ∧ (C1 ∨ C2).
Here is a particular case of the top item with I = {1, 2} and J = {7, 8, 9}.
(A1,7 ∩A1,8 ∩A1,9)∪(A2,7 ∩A2,8 ∩A2,9) ⊆ (A1,7 ∪A2,7)∩(A1,8 ∪A2,8)∩(A1,9 ∪A2,9)
which can be easier to read when written as:
(A1 ∩ B1 ∩ C1) ∪ (A2 ∩ B2 ∩ C2) ⊆ (A1 ∪ A2) ∩ (B1 ∪ B2) ∩ (C1 ∪ C2).
Exercise: Show that converse implications and reverse inclusions do not hold.