Bocconi Mathematics 30062 · exam prep

Bocconi Mathematics 30062:
every formula, one search away.

1271 entries from the Bocconi Mathematics 30062 Module 1 (General) lectures and the course textbook — every definition, formula, theorem and proof — each with a plain-English explanation and a breakdown of what every symbol stands for. Built to help you prepare efficiently and drill until it sticks.

Formulas: 1271
Topics: 22
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1271 results

Sets & set operations35

Real numbers & completeness5

Symbol index

Every symbol used in the sheet, and what it stands for.

$A, B$ sets, denoted by capital letters
$x, a$ elements, denoted by lowercase letters
${\cdot}$ set notation
$x ∣ P (x)$ the set of x satisfying property P
$\in$ is an element of
$\in /$ is not an element of
$\subseteq$ subset or equal
$\subset$ proper subset
$\exists$ there exists
$⊈$ is not a subset of
$A = B$ set equality
$\emptyset$ the empty set
$\cup$ union operator
$\cap$ intersection operator
$A \cap B = \emptyset$ condition for disjointness
$⋃_{i = 1}^{N} A_{i}$ union over the family of sets A_1,...,A_N
$⋂_{i = 1}^{N} A_{i}$ intersection over the family of sets A_1,...,A_N
$∖$ set difference operator
$\cup, \cap$ union and intersection
$U$ the universal set
$A^{c}$ complement of A
$A ∖ B$ set difference
$B^{c}$ complement of B
$(A^{c})^{c}$ complement of the complement of A
$(A \cap B)^{c}$ complement of the intersection
$A^{c} \cup B^{c}$ union of the complements
$N$ set of natural numbers
$Z$ set of integers
$Q$ set of rational numbers
$R$ set of real numbers
$Q^{c}$ irrational numbers
$p, q, r$ real/rational numbers
$>, <, =$ order relations
$\geq, \leq$ weak order relations
$A$ subset of R
${z ∣ x \leq z \leq y}$ all points between x and y
$[x, y]$ closed interval
$(x, y)$ open interval
$[x, y), (x, y]$ half-open intervals
$\infty, - \infty$ positive/negative infinity, used as unbounded endpoints
$R$ extended real line
$h$ candidate bound
$max A, min A$ maximum and minimum of A
$sup A$ least upper bound of A
$in f A$ greatest lower bound of A
$sup A, max A$ supremum and maximum of A
$in f A, min A$ infimum and minimum of A
$∣ x ∣$ absolute value of x
$∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ triangle inequality
$c$ positive constant
$d (x, y)$ distance between x and y
$B_{ϵ} (x_{0})$ neighborhood of x_0 with radius epsilon
$ϵ$ radius
$x_{0}$ center point
$B_{ϵ}^{+} (x_{0})$ right half-neighborhood
$B_{ϵ}^{-} (x_{0})$ left half-neighborhood
$B_{x_{0}} (\infty), B_{x_{0}} (- \infty)$ one-sided infinite neighborhoods
$B_{ϵ}^{-} (x)$ left neighborhood of x with radius epsilon
$B_{ϵ}^{+} (x)$ right neighborhood of x with radius epsilon
$\times$ Cartesian product operator
$(a_{1}, a_{2})$ ordered pair
$R^{2}$ the Cartesian plane
$(12)$ column vector representation of the point (1,2)
$R^{3}$ three-dimensional Cartesian space
$(x, y, z)$ ordered triple
$R^{n}$ n-dimensional Cartesian space
$(x_{1}, \dots, x_{n})$ ordered n-tuple
$x_{1}, \dots, x_{n}$ components of the vector
$x, y \in R^{n}$ vectors of the same dimension n
$x_{i}, y_{i}$ i-th components of x and y
$u, v$ two vectors sharing a starting point
$x, y, z \in R^{n}$ arbitrary vectors
$0$ zero vector
$- x$ additive inverse of x
$α \in R$ scalar
$x \in R^{n}$ vector
$x, y \in R^{n}$ vectors
$α, β \in R$ scalars
$α_{i}$ scalar coefficients
$x_{i}$ vectors being combined
$x \cdot y$ resulting scalar
$x ⊥ y$ x is orthogonal to y
$α \in [0, 1]$ weight parameter
$α_{i} \geq 0$ non-negative weights
$\sum α_{i} = 1$ weights sum to one
$A \subseteq R^{n}$ the set in question
$∥ x ∥$ length (norm) of x
$ϵ > 0$ radius
$x_{0} \in R^{n}$ center point
$d (x, x_{0})$ Euclidean distance from x to x0
$int A$ set of interior points of A
$ext A$ set of exterior points of A
$\partial A$ set of boundary points of A
$A = cl A$ closure of A
$A^{'}$ derived set of A (set of all limit points)
$int A$ interior of A
$B_{ϵ} (x)$ open ball of radius ϵ around x
$K$ a fixed positive radius bounding all points of A
$B_{K} (0)$ ball of radius K centered at the origin containing A
$δ$ radius of the smaller ball around y, chosen as ϵ − ‖x−y‖
$B$ codomain of f
$f (x) = y$ f associates y ∈ B to x ∈ A
$x$ independent variable / input
$y = f (x)$ dependent variable / output, the image of x
$Im (f)$ range / image of f
$Gr (f)$ graph of f
$m$ slope
$b$ y-intercept
$a, b, c$ real coefficients, a ≠ 0
$k$ proportionality constant
$α$ exponent, α ∈ ℚ
$a$ vertical shift amount
$f$ inner function, f: A → B
$g$ outer function, g: C → D, with f(A) ⊆ C
$f^{- 1}$ inverse function of f
$f : A \subseteq R \to R$ the function under consideration
$Im (f)$ the image (range) of f
$f : R^{n} \to R$ a multivariable real-valued function
$sup_{x \in A} f (x)$ least upper bound of the function's values on A
$in f_{x \in A} f (x)$ greatest lower bound of the function's values on A
$e^{2}$ supremum, attained at x = 2
$f increasing$ hypothesis: x>y \Rightarrow f(x)\ge f(y)
$max_{x \in A} f (x)$ the maximum of Im(f), i.e. the global maximum value
$C$ the set over which we optimize
$arg max$ set of maximizers
$K, L$ capital and labor inputs
$p_{K}, p_{L}$ prices of capital and labor
$λ$ Lagrange multiplier on the budget constraint
$w, c$ quantities of wine and cheese consumed
$B_{ϵ}^{+} (x)$ right-neighborhood of x of radius epsilon
$B_{δ}^{-} (x)$ left-neighborhood of x of radius delta
$α f (x) + (1 - α) f (y)$ value of the chord (line segment) at that point
$epi (f)$ the set of points lying on or above the graph of f
$hypo (f)$ the set of points lying on or below the graph of f
$g_{i}$ single-variable function acting only on coordinate x_i
$x = (x_{1}, \dots, x_{n})$ point in the domain A
$f (x)$ global maximum/minimum value
$C \subseteq A$ subset of the domain over which we optimize
$α x + (1 - α) y$ convex combination of x and y
$ω_{i} \geq 0$ non-negative weights
$f^{- 1} (k)$ preimage of the value k, i.e. the level set
$u$ utility function
$(u \geq k)$ bundles delivering at least utility level k
$p_{x}, p_{y}$ prices of the two goods
$w$ consumer's wealth/budget
$g \circ f$ composition of increasing transformation g with f
$x_{n}$ the n-th term of the sequence
$f : N \to R$ sequence viewed as a function
$Im (f)$ range: the set of terms
$L$ the limit of the sequence

Sets & set operations35

What is a Set?

Ways to Describe a Set

Set Membership

Subset

Proper Subset

Not a Subset

Set Equality

Empty Set

Union

Intersection

Disjoint Sets

Union and Intersection of Families of Sets

Set Difference

Commutative Property of Union and Intersection

Associative Property of Union and Intersection

Distributive Property of Set Operations

Universal Set

Complement

Set Difference via Complement

Double Complement

De Morgan's Laws

Proof Sketch of De Morgan's Law (A∩B)ᶜ = Aᶜ∪Bᶜ

Natural Numbers

Integers

Rational Numbers

Proof that √2 is Irrational

The Real Numbers

Hierarchy of Number Sets

Density and Continuity of Q and R

The Natural Order on R

R² Has No Natural Total Order

Interval

Types of Bounded Intervals

Unbounded Intervals

The Extended Real Line

Real numbers & completeness5

Upper and Lower Bound

Bounded Above / Below / Bounded

Maximum and Minimum

Uniqueness of Maximum and Minimum

Supremum and Infimum

Symbol index