Set Theory & Venn Diagrams

Cards · 20

Union of sets A and B (A ∪ B)

{x | x ∈ A or x ∈ B}; the set of all elements that belong to set A, set B, or both.

Intersection of sets A and B (A ∩ B)

{x | x ∈ A and x ∈ B}; the set of all elements that belong simultaneously to both set A and set B.

Set Difference / Relative Complement (A \ B or A - B)

{x | x ∈ A and x ∉ B}; the set of elements that belong to set A but do not belong to set B.

Absolute Complement of set A (A' or Aᶜ)

{x ∈ U | x ∉ A}; the set of all elements in the universal set U that are not members of set A.

Symmetric Difference of A and B (A △ B)

{x | (x ∈ A and x ∉ B) or (x ∈ B and x ∉ A)}; the set of elements in either A or B, but not both.

Subset (A ⊆ B)

∀x (x ∈ A → x ∈ B); every element contained in set A is also an element of set B.

Proper Subset (A ⊂ B)

A ⊆ B and A ≠ B; every element of A is in B, and B contains at least one element not in A.

Cartesian Product of sets A and B (A × B)

{(a, b) | a ∈ A and b ∈ B}; the set of all ordered pairs where the first element is from A and the second is from B.

Power Set of set A (P(A))

{S | S ⊆ A}; the set containing all possible subsets of set A, including the empty set and A itself.

Empty Set / Null Set (∅ or {})

{x | x ≠ x}; the unique set that contains no elements, having a cardinality of zero.

Universal Set (U)

{x | x ∈ U}; the overarching set containing all objects and elements under consideration in a given context.

Disjoint Sets

A ∩ B = ∅; two sets that share no common elements, represented as non-overlapping circles in a Venn diagram.

Cardinality of a finite set (|A| or n(A))

The total count of distinct elements contained within set A.

Set Equality (A = B)

∀x (x ∈ A ↔ x ∈ B); sets A and B are equal if and only if A ⊆ B and B ⊆ A.

De Morgan's Law for Union Complement

(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ; the complement of a union equals the intersection of the individual complements.

De Morgan's Law for Intersection Complement

(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ; the complement of an intersection equals the union of the individual complements.

Principle of Inclusion-Exclusion for two sets

|A ∪ B| = |A| + |B| - |A ∩ B|; counts union size by summing sizes and subtracting the overlapping intersection.

Venn Diagram representation of A ∩ B

The mutually overlapping region shared between circle A and circle B.

Venn Diagram representation of A \ B

The region inside circle A strictly excluding the overlapping area shared with circle B.

Venn Diagram representation of (A ∪ B)ᶜ

The entire shaded region inside universal bounding rectangle U that lies outside both circles A and B.