Introduction to set theory and to methodology and philosophy ofmathematics and computer programming

Indexed families of sets

An overview

by Jan Plaza

c©2017 Jan PlazaUse under the Creative Commons Attribution 4.0 International License

Version of March 10, 2017

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 anindexed family of open intervals of reals. We have, X 1

2= (0, 11

2), etc.

If T = {t ∈ N : t < n} we write {Xt}t<n .

If T = {t ∈ Z : k 6 t 6 n} we write {Xt}nt=k .

If T = {t ∈ Z : k 6 t} we write {Xt}∞t=k .

The union of {Xt}t∈T , denoted⋃

t∈T Xt , is⋃{Xt : t ∈ T}

– the union of the corresponding family of sets.

Let T 6=∅. 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 6 t < n} we write⋃

t<n Xt and⋂

t<n Xt (if n > 0).

If T = {t ∈ Z : k 6 t 6 n} we write⋃n

t=k Xt and⋂n

t=k Xt (if n > k).

If T = {t ∈ Z : k 6 t} we write⋃∞

t=k Xt and⋂∞

t=k Xt .

Fact1. x ∈

⋃t∈T Xt iff ∃t∈T x ∈ Xt

2. x ∈⋂

t∈T Xt iff ∀t∈T x ∈ Xt

ExampleLet Xn = (− 1

n , 1 + 1n ), for n ∈ Z+.

This is an indexed family of open intervals of real numbers.

X1 =(−1, 2), X2 =(−12 , 11

2), X3 =(−13 , 11

3), ..., X10 =(− 110 , 1 1

10), ...(−1, 2) ⊃ (−1

2 , 112) ⊃ (−1

3 , 113) ⊃ ... ⊃ (− 1

10 , 1 110) ⊃ ...

⋃t∈Z+ Xn = ?

(−1, 2)

0, 1 ∈ (−1, 2), 0, 1 ∈ (−12 , 11

2), 0, 1 ∈ (−13 , 11

3), ..., 0, 1 ∈ (− 110 , 1 1

10), ...⋂t∈Z+ Xn = [0, 1]

ExampleLet Xn = (− 1

n , 1 + 1n ), for n ∈ Z+.

This is an indexed family of open intervals of real numbers.X1 =(−1, 2), X2 =(−1

2 , 112), X3 =(−1

3 , 113), ..., X10 =(− 1

10 , 1 110), ...

(−1, 2) ⊃ (−12 , 11

2) ⊃ (−13 , 11

3) ⊃ ... ⊃ (− 110 , 1 1

10) ⊃ ...

⋃t∈Z+ Xn = ?

(−1, 2)

0, 1 ∈ (−1, 2), 0, 1 ∈ (−12 , 11

2), 0, 1 ∈ (−13 , 11

3), ..., 0, 1 ∈ (− 110 , 1 1

10), ...⋂t∈Z+ Xn = [0, 1]

ExampleLet Xn = (− 1

n , 1 + 1n ), for n ∈ Z+.

This is an indexed family of open intervals of real numbers.X1 =(−1, 2), X2 =(−1

2 , 112), X3 =(−1

3 , 113), ..., X10 =(− 1

10 , 1 110), ...

(−1, 2) ⊃ (−12 , 11

2) ⊃ (−13 , 11

3) ⊃ ... ⊃ (− 110 , 1 1

10) ⊃ ...⋃t∈Z+ Xn = ?

(−1, 2)

0, 1 ∈ (−1, 2), 0, 1 ∈ (−12 , 11

2), 0, 1 ∈ (−13 , 11

3), ..., 0, 1 ∈ (− 110 , 1 1

10), ...⋂t∈Z+ Xn = [0, 1]

ExampleLet Xn = (− 1

n , 1 + 1n ), for n ∈ Z+.

This is an indexed family of open intervals of real numbers.X1 =(−1, 2), X2 =(−1

2 , 112), X3 =(−1

3 , 113), ..., X10 =(− 1

10 , 1 110), ...

(−1, 2) ⊃ (−12 , 11

2) ⊃ (−13 , 11

3) ⊃ ... ⊃ (− 110 , 1 1

10) ⊃ ...⋃t∈Z+ Xn = (−1, 2)

0, 1 ∈ (−1, 2), 0, 1 ∈ (−12 , 11

2), 0, 1 ∈ (−13 , 11

3), ..., 0, 1 ∈ (− 110 , 1 1

10), ...⋂t∈Z+ Xn = [0, 1]

ExampleLet Xn = (− 1

n , 1 + 1n ), for n ∈ Z+.

This is an indexed family of open intervals of real numbers.X1 =(−1, 2), X2 =(−1

2 , 112), X3 =(−1

3 , 113), ..., X10 =(− 1

10 , 1 110), ...

(−1, 2) ⊃ (−12 , 11

2) ⊃ (−13 , 11

3) ⊃ ... ⊃ (− 110 , 1 1

10) ⊃ ...⋃t∈Z+ Xn = (−1, 2)

0, 1 ∈ (−1, 2), 0, 1 ∈ (−12 , 11

2), 0, 1 ∈ (−13 , 11

3), ..., 0, 1 ∈ (− 110 , 1 1

10), ...

⋂t∈Z+ Xn = ?

[0, 1]

ExampleLet Xn = (− 1

n , 1 + 1n ), for n ∈ Z+.

This is an indexed family of open intervals of real numbers.X1 =(−1, 2), X2 =(−1

2 , 112), X3 =(−1

3 , 113), ..., X10 =(− 1

10 , 1 110), ...

(−1, 2) ⊃ (−12 , 11

2) ⊃ (−13 , 11

3) ⊃ ... ⊃ (− 110 , 1 1

10) ⊃ ...⋃t∈Z+ Xn = (−1, 2)

0, 1 ∈ (−1, 2), 0, 1 ∈ (−12 , 11

2), 0, 1 ∈ (−13 , 11

3), ..., 0, 1 ∈ (− 110 , 1 1

10), ...⋂t∈Z+ Xn = ?

[0, 1]

ExampleLet Xn = (− 1

n , 1 + 1n ), for n ∈ Z+.

This is an indexed family of open intervals of real numbers.X1 =(−1, 2), X2 =(−1

2 , 112), X3 =(−1

3 , 113), ..., X10 =(− 1

10 , 1 110), ...

(−1, 2) ⊃ (−12 , 11

2) ⊃ (−13 , 11

3) ⊃ ... ⊃ (− 110 , 1 1

10) ⊃ ...⋃t∈Z+ Xn = (−1, 2)

0, 1 ∈ (−1, 2), 0, 1 ∈ (−12 , 11

2), 0, 1 ∈ (−13 , 11

3), ..., 0, 1 ∈ (− 110 , 1 1

10), ...⋂t∈Z+ Xn = [0, 1]

Unions and intersections of indexed subfamilies

Let T0 ⊆ T . Then, {Xt}t∈T0 is an indexed subfamily of {Xt}t∈T .

FactLet T0 ⊆ T . Then:1.

⋃t∈T0

Xt ⊆⋃

t∈T Xt

2.⋂

t∈T0Xt ⊇

⋂t∈T Xt

Exercise1. Disprove: if T0 ⊂ T then

⋃t∈T0

Xt ⊂⋃

t∈T Xt

2. Disprove: if T0 ⊂ T then⋂

t∈T0Xt ⊃

⋂t∈T Xt

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 6=∅.

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 6=∅.

So, families of sets and indexed families of sets can be treated interchangingly.

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 ∧, ∩.

PropositionAi ⊆ B, for every i ∈ I iff

⋃i∈I Ai ⊆ B.

Ai ⊇ B, for every i ∈ I iff⋂

i∈I Ai ⊇ B (assuming I 6=∅).

Analogous statements

I ∀v (A(v)→ B)⇔ (∃v A(v))→ B (assuming v 6∈ var(B)).∀v (A(v)← B)⇔ (∀v A(v))← B (assuming v 6∈ var(B)).

I∧∧

i∈I (Ai → B)⇔ (∨∨

i∈I Ai )→ B.∧∧i∈I (Ai ← B)⇔ (

∧∧i∈I Ai )← B.

I 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.

I 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.

PropositionAi0 ⊆

⋃i∈I Ai , for any index i0 ∈ I .

Ai0 ⊇⋂

i∈I Ai , for any index i0 ∈ I (assuming I 6=∅).

Analogous statements

I A[ tv ]⇒ ∃v A , for any term t.A[ tv ]⇐ ∀v A , for any term t.Holds because the universe is assumed to be non-empty.

I Ai0 ⇒∨∨

i∈I Ai , for any index i0 ∈ I .Ai0 ⇐

∧∧i∈I Ai , for any index i0 ∈ I .

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

I 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

Proposition⋂i∈I Ai ⊆

⋃i∈I Ai (assuming I 6=∅).

Analogous statements

I ∀v A⇒ ∃v A.It holds because the universe of quantification is always assumed to be non-empty.

I∧∧

i∈I Ai ⇒∨∨

i∈I Ai

I A particular case of the item above with I = {1, 2}:A1 ∧ A2 ⇒ A1 ∨ A2

I A particular case of the Proposition above, with I = {1, 2}:A1 ∩ A2 ⊆ A1 ∪ A2

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

I ¬∀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∈I Ai ⇔∨∨

i∈I ¬Ai .¬∨∨

i∈I Ai ⇔∧∧

i∈I ¬Ai .

I A particular case of the item above with I = {1, 2}:¬(A1 ∧ A2)⇔ ¬A1 ∨ ¬A2.¬(A1 ∨ A2)⇔ ¬A1 ∧ ¬A2.

I A particular case of the Proposition above, with I = {1, 2}:(A1 ∩ A2)c = A1

c ∪ A2c .

(A1 ∪ A2)c = A1c ∩ A2

c .

Proposition [distributivity laws 1]

1.⋃

i∈I (A ∪ Bi ) = A ∪ (⋃

i∈I Bi ).⋂i∈I (A ∩ Bi ) = A ∩ (

⋂i∈I Bi ) (assuming I 6=∅).

2.⋃

i∈I (A ∩ Bi ) = A ∩ (⋃

i∈I Bi ).⋂i∈I (A ∪ Bi ) = A ∪ (

⋂i∈I Bi ) (assuming I 6=∅).

Statements analogous to item 2.I ∃v (A ∧ B(v))⇔ A ∧ ∃v B(v) (assuming v 6∈ var(A)).∀v (A ∨ B(v))⇔ A ∨ ∀v B(v) (assuming v 6∈ var(A)).

I∨∨

i∈I (A ∧ Bi )⇔ A ∧ (∨∨

i∈I Bi ).∧∧i∈I (A ∨ Bi )⇔ A ∨ (

∧∧i∈I Bi ).

I 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).

I 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.

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 6=∅)

2.⋃

i∈I (Ai ∩ Bi ) ⊆ (⋃

i∈I Ai ) ∩ (⋃

i∈I Bi ).⋂i∈I (Ai ∪ Bi ) ⊇ (

⋂i∈I Ai ) ∪ (

⋂i∈I Bi ) (assuming I 6=∅)

Statements analogous to item 2.I ∃v (A ∧ B)⇒ ∃v A ∧ ∃v B.∀v (A ∨ B)⇐ ∀v A ∨ ∀v B.

I∨∨

i∈I (Ai ∧ Bi )⇒ (∨∨

i∈I Ai ) ∧ (∨∨

i∈I Bi ).∧∧i∈I (Ai ∨ Bi )⇐ (

∧∧i∈I Ai ) ∨ (

∧∧i∈I Bi ).

I 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).

I 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.

The remaining slides present an extra credit material.

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.

Every doubly indexed family of sets over S and T can be viewed asa 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 indexis 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 6=∅. 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 .

Also, every doubly indexed family of sets over S and T can be viewed asa 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 indexis 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 6=∅. 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 .

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 S1s = 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 .

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 S1s = 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 iffx ∈ T7 ∩ T8 ∩ T9 iffx ∈ (X1,7∪,X2,7) ∩ (X1,8∪,X2,8) ∩ (X1,9∪,X2,9) iff∀t∈T ∃s∈S x ∈ Xs,t .

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 6=∅)4. x ∈

⋂t∈T

⋂s∈S Xs,t iff ∀t∈T ∀s∈S x ∈ Xs,t (assuming S ,T 6=∅)

5. x ∈⋃

s∈S⋂

t∈T Xs,t iff ∃s∈S ∀t∈T x ∈ Xs,t (assuming T 6=∅)6. x ∈

⋃t∈T

⋂s∈S Xs,t iff ∃t∈T ∀s∈S x ∈ Xs,t (assuming S 6=∅)

7. x ∈⋂

s∈S⋃

t∈T Xs,t iff ∀s∈S ∃t∈T x ∈ Xs,t (assuming S 6=∅)8. x ∈

⋂t∈T

⋃s∈S Xs,t iff ∀t∈T ∃s∈S x ∈ Xs,t (assuming T 6=∅)

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 6=∅).

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 6=∅).

PropositionLet {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 twobig union or intersection operations.⋃

i∈I⋂

j∈J Ai ,j� � //

⋂j∈J

⋃i∈I Ai ,j� u

''⋂i∈I ,j∈J Ai ,j

)

77

� � //� u

''

⋃i∈I ,j∈J Ai ,j

⋃j∈J

⋂i∈I Ai ,j

� � //⋂

i∈I⋃

j∈J Ai ,j

)

77

PropositionLet {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 twobig union or intersection operations.⋃

i∈I⋂

j∈I Ai ,j� � //

� u

((

⋂j∈I

⋃i∈I Ai ,j� t

''⋂i ,j∈I Ai ,j

*

77

� � //� t

''

⋂i∈I Ai ,i

)

66

� � //� u

((

⋃i∈I Ai ,i

� � //⋃

i ,j∈I Ai ,j

⋃j∈I

⋂i∈I Ai ,j

)

66

� � //⋂

i∈I⋃

j∈I Ai ,j

*

77

An analogy with quantifiers

⋃i∈I

⋂j∈I Ai ,j

� � //� u

((

⋂j∈I

⋃i∈I Ai ,j� t

''⋂i ,j∈I Ai ,j

*

77

� � //� t

''

⋂i∈I Ai ,i

)

66

� � //� u

((

⋃i∈I Ai ,i

� � //⋃

i ,j∈I Ai ,j

⋃j∈I

⋂i∈I Ai ,j

)

66

� � //⋂

i∈I⋃

j∈I Ai ,j

*

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:

Analogies explored

Conisder the inclusion and the implication on the top of the two diagrams.

I⋃

i∈I⋂

j∈J Ai ,j ⊆⋂

j∈J⋃

i∈I Ai ,j

I ∃v ∀w A(v ,w)⇒ ∀w ∃v A(v ,w)

I∨∨

i∈I∧∧

j∈J Ai ,j ⇒∧∧

j∈J∨∨

i∈I Ai ,j

I 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).

I 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.