Set Theory

Set

A collection of objects. Each object in the collection is called an element of the set.

Element

Each individual object belonging to a set. If a is an element of set A, we write a ∈ A.

Roster/Explicit Notation

A way of writing a set by listing all its elements explicitly inside curly braces. Example: A = {1, 2, 3, 4}

Set-Builder Notation

A way of writing a set by describing a property its elements must satisfy. Example: B = {x

a ∈ A

Notation meaning “a IS an element of set A.”

a ∉ A

Notation meaning “a is NOT an element of set A.”

Empty Set (∅)

A set that contains no elements at all, also called the null or void set. Denoted ∅ or {}.

ℕ (Natural Numbers)

The set of positive integers: ℕ = {1, 2, 3, …}. Sometimes includes 0 (written ℕ*).

ℤ (Integers)

The set of all integers: {…, −2, −1, 0, 1, 2, …}

ℚ (Rational Numbers)

The set of all numbers that can be written as a/b where a, b ∈ ℤ and b ≠ 0.

ℝ (Real Numbers)

The set of all real numbers, including rationals and irrationals.

Subset (A ⊆ B)

A is a subset of B (written weakly) if every element of A is also an element of B. i.e., x ∈ A ⟹ x ∈ B.

Proper Subset (A ⊂ B)

A is a proper subset of B if A ⊆ B AND there exists at least one element y ∈ B such that y ∉ A. A ≠ B.

Equal Sets (A = B)

Two sets A and B are equal if A ⊆ B AND B ⊆ A; that is, they contain exactly the same elements.

Cardinality

A

Countably Infinite Set

A set whose elements can be put in one-to-one correspondence with the natural numbers ℕ. Cardinality = ℵ₀ (Aleph null).

Continuum / Uncountably Infinite

A set like an interval [a, b] on the real line whose cardinality is greater than ℵ₀; denoted by c.

Finite Set

A set with a limited, countable number of elements.

Infinite Set

A set with an unlimited number of elements. Example: B = {x ∈ ℝ : 0 ≤ x ≤ 53} is an infinite set.

Interval [a, b]

The closed interval: {x ∈ ℝ : a ≤ x ≤ b}. Both endpoints a and b are included. Uses square brackets [ ].

Interval (a, b)

The open interval: {x ∈ ℝ : a < x < b}. Neither endpoint is included. Uses parentheses ( ).

Interval [a, b)

The half-open interval: {x ∈ ℝ : a ≤ x < b}. Left endpoint included, right endpoint excluded.

Interval (a, b]

The half-open interval: {x ∈ ℝ : a < x ≤ b}. Left endpoint excluded, right endpoint included.

Interval (a, ∞)

The set {x ∈ ℝ : x > a}. All real numbers strictly greater than a. Infinity always uses parenthesis.

Interval (−∞, b]

The set {x ∈ ℝ : x ≤ b}. All real numbers less than or equal to b.

Set Intersection (A ∩ B)

The set of all elements that belong to BOTH A and B. A ∩ B = {x

Set Union (A ∪ B)

The set of all elements that belong to A OR B (or both). A ∪ B = {x

Set Difference (A \ B)

The set of elements in A that are NOT in B. A \ B = {x

Complement of A (A’)

The set of all elements in the universal set U that are NOT in A. A’ = U \ A. Properties: A ∪ A’ = U; A ∩ A’ = ∅.

Universal Set (U)

The set containing all elements under consideration in a particular context. All sets in the discussion are subsets of U.

Idempotent Laws

A ∪ A = A and A ∩ A = A. A set unioned or intersected with itself is just itself.

Associative Laws

( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) and ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ). Grouping does not matter for union or intersection.

Commutative Laws

A ∪ B = B ∪ A and A ∩ B = B ∩ A. Order does not matter for union or intersection.

Distributive Laws

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

Identity Laws

A ∪ ∅ = A; A ∩ U = A; A ∪ U = U; A ∩ ∅ = ∅. Union with empty set or intersection with universal set leaves A unchanged.

Complement Laws

A ∪ A’ = U; A ∩ A’ = ∅; (A’)’ = A; U’ = ∅; ∅’ = U.

De Morgan’s Laws

(A ∪ B)’ = A’ ∩ B’ and (A ∩ B)’ = A’ ∪ B’. The complement of a union is the intersection of complements, and vice versa.

Absorption Laws

A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A. Absorbing a subset does not change the set.

Theorem — Equivalent Conditions for A ⊆ B

Each of the following is equivalent to A ⊆ B: (1) A ∩ B = A, (2) A ∪ B = B, (3) B’ ⊆ A’, (4) A ∩ B’ = ∅, (5) B ∪ A’ = U.

Two-Set Venn Diagram Regions

For sets A and B in U, the four disjoint regions are: A∩B (both), A∩B’ (only A), A’∩B (only B), A’∩B’ (neither).

Inclusion-Exclusion Principle (2 sets)

n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Used to find the size of the union without double-counting the intersection.

Three-Set Inclusion-Exclusion

n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(A∩C) − n(B∩C) + n(A∩B∩C).

Absolute Value Definition

Absolute Value Property — Equation

If

Absolute Value Property —

xy

Absolute Value Inequality — Less Than

If

Absolute Value Inequality — Greater Than

If

Quadratic Equation

An equation of the form ax² + bx + c = 0 where a, b, c ∈ ℝ and a ≠ 0.

Quadratic Formula

For ax² + bx + c = 0, the solutions are x = [−b ± √(b²−4ac)] / (2a).

Discriminant (b²−4ac)

The expression b²−4ac in the quadratic formula. If > 0: two distinct real roots. If = 0: one repeated real root. If < 0: two complex conjugate roots.

Logarithmic Function

y = log_b(x) where b > 0, b ≠ 1, x > 0. Defined as: y = log_b(x) if and only if x = b^y.

Log Property 1 — Product Rule

log_b(AB) = log_b(A) + log_b(B). The log of a product equals the sum of the logs.

Log Property 2 — Quotient Rule

log_b(A/B) = log_b(A) − log_b(B). The log of a quotient equals the difference of the logs.

Log Property 3 — log_b(b) = 1

log_b(b) = 1, since b¹ = b. The log base b of b itself is always 1.

Log Property 4 — log_b(1) = 0

log_b(1) = 0, since b⁰ = 1. The log of 1 is always 0 for any valid base.

Log Property 5 — log_b(b^x) = x

log_b(b^x) = x. A logarithm undoes the exponential with the same base.

Log Property 6 — b^(log_b x) = x

b^(log_b x) = x. An exponential undoes the logarithm with the same base.

Common Logarithm (log)

Logarithm with base 10; written as log₁₀ or simply log. Used for base-10 exponential equations.

Natural Logarithm (ln)

Logarithm with base e ≈ 2.718; written ln(x) = log_e(x). Used in growth/decay and calculus applications.

Change of Subject using Logarithms

To make t the subject of y = (1+r)^t: take log of both sides → log y = t · log(1+r) → t = log(y) / log(1+r).