Implementation of Computational Methods

Device Interconnection

Table of contents:

• Set Theory
  • Sets
    • Explicit Sets
    • Implicit Sets
    • Empty Set
    • Power Set
  • Symbols
    • Set Cardinality
    • Equality
    • Relation: Membership
    • Subsets
    • Proper Subsets
    • Union
    • Intersection
    • Difference
    • Complement

Set Theory

Sets

Explicit Sets

• A = {3, 6, 9, 12}
• B = {Juan, Luis, Pedro}
• C = {0, 1}

Implicit Sets

Symbols:

∋/| = such that

∧ = and

V = or

Examples:

• D = {x ∋ x/2 = 0 }
• E = {x ∋ x is a planet}
• F = {x | x is natural number ∧ x es even}

Empty Set

{} = ∅

Power Set

For set A, its power set is the set whose elements are each of the possible subsets of A.

A = {a, b, c}

P(A) = {∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}

Symbols

Set Cardinality

The number of elements in a set:

A = {2, 5, 7, 9}

|A| = 4

Cardinality is always a natural number.

Equality

Equal sets have the same elements:

A = {2, 5, 7, 9}

B = {5, 2, 9, 7}

A = B

Relation: Membership

a belongs to A:

a ∈ A

b does not belong to A: b ∉ A

Examples:

'a' ∈ {'a', 'e', 'i', 'o', 'u'}

0 ∉ {1, 2, 3, 4}

Subsets

A subset is a set that contains only elements that are also in another set:

B is a subset of A: B ⊆ A

C is not a subset of A: C ⊈ A

Proper Subsets

A proper subset is a subset that contains some, but not all, of the elements of another set:

C is a proper subset of A: C ⊂ A

D is not a proper subset of A: D ⊄ A

Union

A ∪ B = {x ∍ x ∈ A ∨ x ∈ B}

Example:

A = {1, 3, 5}

B= {2, 4, 5, 6}

A∪B={1, 2, 3, 4, 5, 6}

Properties:

• Idempotence: A∪A = A

• Commutativity: A∪B = B∪A

• Associativity: (A∪B)∪C = A∪(B∪C)

• Identity: A∪U = U

• Null: A∪∅ = A

Intersection

A ∩ B = {x ∍ x∈A ∧ x∈B}

Example:

A = {1, 3, 5}

B = {2, 4, 5, 6}

A ∩ B = {5}

Properties:

• Idempotence: A ∩ A = A
• Commutativity: A ∩ B = B ∩ A
• Associativity: (A∩B)∩C = A∩(B∩C)
• Identity: A ∩ U = A
• Neutral: A ∩ ∅ = ∅

Difference

A-B = {x ∍ x∈A ∧ x∉B}

Example:

A = {1, 3, 5}

B = {2, 4, 5, 6}

A-B = {1, 3}

B-A = {2, 4, 6}

Complement

A^c = {x ∍ x∈U ∧ x∉A}

Example:

U = ℤ

A = {x ∍ x is even}

A^c = {x ∍ x is odd}