MTH 110 Discrete Mathematics I - Basic Set Theory (Vocabulary Flashcards)

Vocabulary flashcards covering key set theory concepts from the notes.

Set

A collection of definite, distinct objects where membership is the only property; order and repetition do not matter.

Element (member)

An object that belongs to a set; written as x ∈ S.

Cardinality

The number of elements in a set, denoted |S|.

Equality of Sets

Two sets S and T are equal if they contain exactly the same elements: ∀x, x ∈ S ⇔ x ∈ T.

Subset

A ⊆ B if every element of A is also an element of B.

Proper Subset

B ⊊ A if B ⊆ A and B ≠ A (B is a subset of A that is not equal to A).

Empty Set

The set with no elements; denoted ∅; ∅ ⊆ A for any A; ∅ is unique.

Universe

A universal set U containing all objects under consideration; operations like complement depend on U.

Union

A ∪ B is the set of elements that are in A or in B (or both).

Intersection

A ∩ B is the set of elements that are in both A and B.

Complement

A^c (relative to U) is the set of elements of U that are not in A.

Difference

A \ B is the set of elements in A but not in B; equivalently A ∩ B^c.

Real Numbers (R)

The set of all real numbers (numbers on the real number line).

Positive Real Numbers

R+ = {x ∈ R | x > 0}.

Natural Numbers

N = {0, 1, 2, 3, …}.

Positive Naturals

N+ = {1, 2, 3, …}.

Integers

Z = {…, -2, -1, 0, 1, 2, …}.

Rational Numbers

Q = {p/q | p,q ∈ Z, q ≠ 0}.

Complex Numbers

C = {x + iy | x,y ∈ R}.

Closed Interval

[a,b] = {x ∈ R | a ≤ x ≤ b} (all numbers between a and b including endpoints).

Open Interval

(a,b] = {x ∈ R | a < x ≤ b} or (a,b) = {x ∈ R | a < x < b} (endpoints excluded accordingly).

Half-Open Interval

[a,b) or (a,b] (one endpoint included, the other excluded).

Singleton

A set with one element: {x}.

Doubleton

A set with two distinct elements: {x, y} with x ≠ y.

Predicate

A statement about a variable x that can be true or false, used in set-builder notation P(x).

Set-Builder Notation

T = {x ∈ S | P(x)} denotes the subset of S consisting of elements that satisfy P.