Set Theory Symbols

member

(x ∈ B; x ∉ B)

• A member is an individual in a set, when a set contains two or more elements. They will normally all be the same kind of thing. Some sets can have an infinite number of members.

empty set

(Ø)

• A set that has no members. This set is not nothing, but just a set with one member of empty set.

universal set

(U)

• A universal set is a set that contains all elements under consideration in the current problem. That is, the universal discourse of a problem.

ordered pair

(〈x,y〉)

• An ordered pair is a way of defining elements in a set in the order in which they occur. They differ from typical sets as the order of the elements does matter.

subset/ set inclusion

(A ⊆ B)

• B is a subset of A if all of the members of B can be found in the universal set A. 

proper subset

(A ⊂ B)

• Used to specify that A is a subset of B, but that they are not equal. A would be entirely found in set B, but set B contains more elements then found in set A.

intersection

(A ∩ B)

• An intersection is a set that contains all of the elements that can be found in a set A and set B.

union

(A ∪ B)

• The union of two sets is the set that contains all elements that are either found in A or in B, but it doesn't have to be all of them.

complement

(A')

• A complement is the set which contains all the elements of a universal set that are not elements of A.

relative complement

(B – A)

• A relative complement is the set of elements that exist in A but not in B.

cardinality

(|B|)

• The cardinality is the number of members that belong in a set.

All students are brilliant.

𝑆 ⊆ B

No students are brilliant.

𝑆 ∩ 𝐵 = ∅

Some students are brilliant.

|𝑆 ∩ 𝐵| ≥ 2

A/Some student is brilliant.

𝑆 ∩ 𝐵 ≠ ∅; or: |𝑆 ∩ 𝐵| ≥ 1

Four students are brilliant.

|𝑆 ∩ 𝐵| = 4

Most students are brilliant.

|𝑆 ∩ 𝐵| > |𝑆 − 𝐵|

Few students are brilliant.

|𝑆 ∩ 𝐵| < some contextually defined number

Both students are brilliant.

𝑆 ⊆ 𝐵 ∧ |𝑆| = 2

Subsective Adjectives

[[adj N]] ⊆ [[N]]

• typical politician

intersective adjectives

[[Adj N]] = [Adj] ∩ [N]

• red apple
  

non-subsective

[[Adj, N]] not ⊆ [N]

• something that may or may not be in the denotation set of the N, ex. alleged collaborator

privative

[Adj N] ∩ [N] = ∅

• counterfeit passport