Membership Relation

Membership relations associate an object and a set, indicating whether the object belongs to the set or not.

This implies there are two possible membership relations:

1. An object a that belongs to a set A is denoted by a ∈ A

2. An object b that doesn't belong to a set A is denoted by b ∉ A

By default, the two relations are indicated by the symbols ∈ and ∉, which require that an object (potential set member) is placed on the left with a set on the right. As a convenince, two additional symbols, ∋ and ∌, are available when this default placement is reversed.

x,y : Obj
A : Set
(the ':' can be considered as is-a property)

x ∈ A
y ∉ A
A ∋ x
A ∌ y

Every object wants to have a meaningful sense of belonging to some set, the blissful period in an object's life when a young objects transitions from objecthood to elementhood, becoming a hooded member.

A membership relation is unlike an inclusion relation; the latter associates one set to another set, establishing by how much is one knee-deep in the other.

Membership relation is reflexive and anti-transitive. In this regard it is unlike inclusion relation, which is transitive.

Inclusion relations

• subset

  • reflexive: A ⊆ A

  • transitive: (A ⊆ B ⋀ B ⊆ C) => A ⊆ C

• proper subset

  • irreflexive:A /⊂ A

A_𝓡_B a𝓡b

(A,A) ∈ 𝓡 (A,B) ∈ 𝓡

a ∈ A R = a ~ A

∧ b ∈ B

if a ∈ A ∧ b ∈ B then it is a ∈ C

If the predicate Φ means "belongs to a set A", then the fact that a ∈ A can also be stated as Φ(a). We can express something about all elements of a set A by begining the formula with ∀a.Φ(a).