Mathematics for Economists - Lecture 1: One-variable Functions

Comprehensive vocabulary flashcards based on Lecture 1 (One-variable functions) covering set theory, elementary functions, functional representations, and introductory economic applications.

Mathematical model

An approximate description or representation of a real-life situation expressed in terms of mathematical symbols.

Set

A collection of distinct objects called elements.

Enumeration

A method of specifying a set by listing all of its individual elements, denoted as $A = \{a_1, a_2, \dots, a_n\}$.

Description (Set Builder Notation)

A method of presenting a set by stating the defining property of its elements, denoted as $A = \{a \mid \text{property}\}$.

Equal sets

Sets that contain exactly the same elements.

Subset

A set $A$ is considered a subset of set $B$ if all elements belonging to $A$ are present in $B$.

Union ($A \cup B$)

The set of all elements that belong to set $A$ or set $B$, defined as $A \cup B = \{x \mid x \in A \text{ or } x \in B\}$, with cardinality formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.

Intersection ($A \cap B$)

The set of all elements that belong to both set $A$ and set $B$, defined as $A \cap B = \{x \mid x \in A \text{ and } x \in B\}$, with cardinality formula $n(A \cap B) = n(A) + n(B) - n(A \cup B)$.

Complement of a set ($A^C$)

The set of elements in the universal set $U$ that do not belong to $A$, defined as $A^C = \{x \mid x \in U \text{ and } x \notin A\}$, with cardinality formula $n(A^C) = n(U) - n(A)$.

Set difference ($A \setminus B$)

The set of elements belonging to $A$ but not to $B$, expressed as $A \setminus B = \{x \mid x \in A \text{ and } x \notin B\}$, with cardinality formula $n(A \setminus B) = n(A) - n(A \cap B)$.

Cartesian product ($A \times B$)

The set of all ordered pairs where the first element is from set $A$ and the second element is from set $B$, given by $A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}$.

Commutative law (of set operations)

The set operation principle stating that $A \cup B = B \cup A$ and $A \cap B = B \cap A$.

Associative law (of set operations)

The set operation principle stating that $A \cup (B \cup C) = (A \cup B) \cup C$ and $A \cap (B \cap C) = (A \cap B) \cap C$.

Distributive law (of set operations)

The set operation principle stating that $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

Variable

A quantity that can vary (change) and get any one of a set of values.

Function

A dependence or relationship between variables consisting of a set of inputs (domain), a set of outputs (range), and a rule by which each input determines exactly one output.

Linear function

A function expressed in the form $f(x) = ax + b$, where $a$ and $b$ are constants.

Quadratic function

A function expressed in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$, and $a$, $b$, and $c$ are constants.

Polynomial function of degree $n$

A function defined by $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $a_n, a_{n-1}, \dots, a_1, a_0$ are constants.

Exponential function

A function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

Logarithmic function

A function defined as $f(x) = \log_a(x)$, where $a > 0$, $x > 0$, and $a \neq 1$.

Rational function

A function expressed as the ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

Analytical representation

A way of representing a function as a formula or set of formulas specifying the mathematical operations to perform on $x$ to determine $y$.

Parametrical representation

A way of representing a function where variables $x$ and $y$ are each expressed in terms of a third variable called a parameter $t$ ($x = \varphi(t)$, $y = \psi(t)$).

Graphical representation

A way of representing a function as a set of points plotted on the Cartesian plane.

Tabular representation

A way of representing a function by listing pairs of values for $x$ and $y$ in a table.

Explicit function

A function directly solved for the dependent variable in terms of the independent variable, written as $y = f(x)$.

Implicit function

A function defined by a relationship in which the dependent and independent variables are not isolated, expressed in the form $F(x, y) = 0$.

Inverse function

A function derived from $y = f(x)$ by solving for the independent variable in terms of the dependent variable, denoted $x = f^{-1}(y)$.

Bounded function

A function for which there exists a constant number $M$ such that for every value of $x$, $|f(x)| \leq M$ (or $-M \leq f(x) \leq M$).

Composite function

A function formed by the superposition of elementary functions, denoted as $f(g(x))$.

Linear demand function

An economic relationship given by $Q = f(P) = aP + b$, where the slope coefficient $a < 0$, $P \geq 0$, and $Q \geq 0$.

Linear supply function

An economic relationship given by $Q = f(P) = aP + b$, where the slope coefficient $a > 0$, $P \geq 0$, and $Q \geq 0$.

Total revenue function ($TR$)

An economic function calculating gross income generated from sales, defined by $TR = P \cdot Q$.

Total cost function ($TC$)

An economic function representing the overall cost of production, defined as $TC = TFC + TVC = TFC + VC \cdot Q$, where $TFC$ is total fixed costs and $VC$ is variable cost per unit.

Profit function ($\pi$)

An economic function defined as total revenue minus total cost: $\pi = TR - TC$.