Set Theory - Vocabulary Flashcards (Lecture Notes)

Vocabulary flashcards covering core Set Theory concepts from the lecture notes.

Set

A well-defined collection of distinct objects, considered as an object in its own right.

Element (member) of a set

An object that is a member of a set; notation x ∈ A.

Not an element

An object that is not a member of a set; notation y ∉ A.

Equality of Sets

Two sets A and B are equal if they have exactly the same elements.

Finite Set

A set with a limited number of elements.

Infinite Set

A set with an unlimited number of elements.

Singleton Set

A set with exactly one element.

Subset

A set A is a subset of B if every element of A is also an element of B (A ⊆ B).

Proper Subset

A subset A ⊂ B where A ⊆ B and A ≠ B.

Superset

A set B that contains A; written B ⊇ A.

Equality by Subsets

A = B if and only if A ⊆ B and B ⊆ A.

Universal Set

The set that contains all objects under consideration (denoted U).

Null (Empty) Set

The set that contains no elements (denoted ∅).

Power Set

P(A) is the set of all subsets of A; |P(A)| = 2^|A|.

Cardinality

The number of elements in a set; denoted |A|.

Union

A ∪ B is the set of elements in A or B or both.

Intersection

A ∩ B is the set of elements common to A and B.

Disjoint Sets

Sets A and B with A ∩ B = ∅ (no elements in common).

Relative Complement

A − B is the set of elements in A but not in B.

Absolute Complement

A′ (or A^c) is the set of elements in the universal set U that are not in A.

Venn Diagram

A diagram showing all possible logical relations between a finite collection of sets.

Cartesian Product

A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B.

Ordered Pair

A pair (a, b) where the order of elements matters, used in Cartesian products.

Commutative Law (Union/Intersection)

A ∪ B = B ∪ A and A ∩ B = B ∩ A.

Associative Law (Union/Intersection)

(A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C).

Distributive Law

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Identity Law

A ∪ ∅ = A and A ∩ U = A.

Complement Law

A ∪ A′ = U and A ∩ A′ = ∅.

Double Complement Law

(A′)′ = A.

De Morgan's Laws

(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Idempotent Law

A ∪ A = A and A ∩ A = A.

Domination Law

A ∪ U = U and A ∩ ∅ = ∅.