UNION AND SEPARATION

"A set is a grouping into a whole of distinct and definite objects of our intuition or of our thought" (Cantor).

---

Union
Semantics

To join two closed expressions of the same type (series or parallel, i.e., sequence or set) is to create a new closed expression consisting of the components of both expressions and also of the same type.


Syntax

(x ∪ y) or x∪y // union of x and y


Definition

Union of two sequences:

⟨( (x∪y = (x↓ y↓)) ← (x/sec ∧ y/sec) )⟩

where ⟨( x/sec = ( x↓ )=x )⟩

Union of sets:

⟨( (x∪y = {x↓ y↓}) ← (x/conj ∧ y/conj) )⟩

being ⟨( x/conj = {x↓}=x )⟩


Justification

Union corresponds to the mind's ability to group several entities into another higher-level entity.


Examples

1. ((a b c) ∪ (d e)) // ev. (a b c d e)
   
   
2. (abc ∪ de) // ev. abcde
   
   
3. ({a b c} ∪ {d e}) // ev. {a b c d e}
   
   
4. {a b c}∪{a b}) // ev. {a b c}
   
   
5. ((a+b c+d) ∪ (s t)) // ev. (a+b c+d s t)
   
   
6. ((a+b+) ∪ (c+d)) // ev. (a+b+c+d)
   
   
7. (a ∪ b ∪ c) // ev. (a b c) eq. abc
   
   
8. ({a} ∪ {b} ∪ {c}) // ev. {a b c}
   
   
9. (a/b) ∪ (c/d) // ev. (a/b c/d)

Remarks

• The condition ((( x↓ )=x) is satisfied by the sequences. The condition ({x↓}=x) is satisfied by sets.
  
  
• When the operands are expressions of different types, the result is the same (since the operation cannot be performed). Example:
  
  ({a b c} ∪ (u v)) // self-evaluates
  
  
• The union only affects the first level of the hierarchy of expressions:
  
  {{a b} {c d}} ∪ {e f} // ev. {{a b} {c d} e f}
  
  
• The union operation is generic, going beyond the classical "union of sets", since the union operates on closed expressions in general and not only on sets.
  
  
• The arguments of the union can be atoms. In this case, note that if x is an atom, (x↓ = x). For example,
  
  (a ∪ b) // ev. (a b) ev. ab for (a↓ = a) and (b↓ = b)
(abc ∪ d) // ev. abcd

Properties

1. ⟨( (x∪y ≡ y∪x) ← ({x↓}=x ∧ {y↓}=y) )⟩
   
   There is commutativity only in the case of sets.
   
   
2. Associativity.
   
   ⟨(((x ∪ y) ∪ z) ≡ (x ∪ (y ∪ z)))
   
   Example:
   
   ((a b) ∪ ((c d) ∪ (e f))) // ev. (a b) ∪ (c d e f)) // ev. (a b c d e f))

((a b) ∪ (c d)) ∪ (e f)) // ev. (a b c d) ∪ (e f)) // ev. (a b c d e f)
   
   
3. ⟨( (x∪x = xx) ← (x↓ = x) )⟩ // atom bonding
   
   This expression is equivalent to ⟨( (x∪x = xx) ← (x# = 1) )⟩
   
   
4. ⟨( (x ∪ θ) = x )⟩
   
   
5. ⟨( (θ ∪ x) = x )⟩
   
   
6. ⟨( ((x ∪ x) = x) ← ({x↓}=x) )⟩

Contrary Union: Separation
Semantics

The opposite operation to union is called "separation". Since the union operation is not commutative in the case of sequences, there is separation on the left and on the right.

In the case of sets, left and right separation coincide.


Syntax

(x ∪' y) // right separation
(x '∪ y) // left separation


Definition

• Separation to the right in sequences:
  
  ⟨ ( (x ∪' y) = ( [x\[1...(x# − y#)]]) ) ←
([x\⌊(x# − y# + 1) ... x#)⌋) = y\[(1...y#)]]) ∧
(x/sec ∧ y/sec) ⟩
  
  It is assumed that x# > y#. The x#−y# first elements of x are selected. The condition is that the elements of y match the elements of x.
  
  
• Separation to the left in sequences:
  
  ⟨ ( (x '∪ y) = ( [x}[(x# − y# +1) ... x#)]]) ) ←
([x\⌊(1 ... y#)⌋) = y\⌊(1 ... y#)⌋) ∧
(x/sec ∧ y/sec) ⟩
  
  
• Separation into sets:
  
  ⟨ ( (x ∪' y) = x/[[y↓]=θ] )←(x/conj ∧ y/conj) ⟩
  
  The elements of x that exist in x are removed from x.

Examples with sequences

1. ((a b c d) ∪' (c d)) // ev. (a b)
   then ((a b) ∪ (c d)) ev. (a b c d)
   
   
2. ((a b c d) '∪ (a b)) // ev. (c d)
   then ((a b) ∪ (c d)) ev. (a b c d)
   
   
3. (abcd ∪' cd) // ev. ab
   
   
4. (abcd '∪ cd) // ev. cd

Examples with sets

1. ({a b c d} ∪' {c d}) // ev. {a b}
   
   
2. ({a b c d} ∪' {e f}) // ev. {a b c d}
   
   
3. ({a b c d} ∪' {a b e f}) // ev. {c d}

Properties

1. ⟨( ((x ∪' y) = z) ↔ (x = (z ∪ y)) )⟩ // separation to the right

⟨( (((x '∪ y) = z)) ↔ (x = (y ∪ z)) )⟩ // left separation
   
   
2. ⟨( (x ∪ y) ∪' y) = x )⟩ // by the definition

⟨( (x ∪ y) '∪ x) = y )⟩ // id.
   
   
3. ⟨( (x ∪' y) ≡ (x '∪ y)) ← ({ x↓ }=x) ∧ { y/b>↓ }=y) )⟩
   
   Left and right separation are equivalent in the case of sets.
   
   
4. ⟨( (x ∪' x) = θ )⟩ // so ⟨( θ∪x = x )⟩

⟨( (x '∪ x) = θ )⟩ // so ⟨( x∪θ = x )⟩
   
   
5. ⟨( (x ∪' θ) = x )⟩ // by the definition
⟨( (x '∪ θ) = x )⟩ // id.