MTH1016 Mathematics for Enterprise Computing 1-5

Comprehensive vocabulary flashcards covering logic, set theory, binary and n-ary relations, functions, and economic applications from MTH1016.

Logic

The study of the validity of reasonings or arguments, or the science of correct reasoning.

Logical Argument

A reasoning structure composed of premises (assumptions) and a conclusion, evaluated by whether the truth of the premises guarantees the truth of the conclusion.

Modus Ponens

A standard valid argument form with the structure: (1) If $p$ is true, then $q$ is true; (2) $p$ is true; (3) Therefore, $q$ is true.

Modus Tollens

A standard valid argument form with the structure: (1) If $p$ is true, then $q$ is true; (2) $q$ is false; (3) Therefore, $p$ is false.

Proposition

A declarative statement that is either strictly true ($T$) or false ($F$), but not both.

Atomic Proposition

A basic proposition that is not built up out of any other propositions.

Compound Proposition

A proposition formed by combining two or more propositions using logical operators.

Negation

The statement "not $p$", denoted by $\neg p$ (or $\sim p$), which reverses the truth value of proposition $p$.

Conjunction

The logical statement "$p$ and $q$", denoted by $p \land q$, which evaluates to true only when both propositions $p$ and $q$ are true.

Disjunction

The logical statement "$p$ or $q$", denoted by $p \lor q$, which evaluates to true when at least one of $p$ or $q$ is true.

Implication

The conditional statement "If $p$ then $q$", written as $p \implies q$, which is false only when premise $p$ is true and conclusion $q$ is false.

Bi-implication

The equivalence statement "$p$ if and only if $q$", denoted by $p \iff q$, which is true when both $p$ and $q$ share the same truth value.

Kleene's 3-Valued Logic (3VL)

A three-valued logic system used in databases to handle missing data ($\text{NULL}$), where conditions evaluate to TRUE, FALSE, or NULL, and queries strictly require a TRUE evaluation to include a row.

Set

A collection of items (elements) where ordering does not matter and two sets are equal if and only if they contain the exact same elements.

Empty Set

The unique set that contains no elements, denoted by $\emptyset$.

Natural Numbers ($\mathbb{N}$)

The infinite set of counting numbers $\mathbb{N} = \{1, 2, 3, \dots\}$, which is closed under addition and multiplication, but not subtraction.

Integers ($\mathbb{Z}$)

The set $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$, containing all natural numbers, zero, and negative whole numbers; it is closed under addition, subtraction, and multiplication, but not division.

Rational Numbers ($\mathbb{Q}$)

The set of all fractions $\mathbb{Q} = \{\frac{m}{n} \mid m \in \mathbb{Z}, n \in \mathbb{N}\}$, which is closed under addition, subtraction, multiplication, and division by non-zero numbers.

Real Numbers ($\mathbb{R}$)

The set containing all numbers on the continuous number line, including all rational numbers as well as irrational numbers like $\sqrt{2}$, $\pi$, and $e$.

Complex Numbers ($\mathbb{C}$)

The set defined as $\mathbb{C} = \{x + i \cdot y \mid x, y \in \mathbb{R}\}$, where $i$ is the imaginary unit satisfying $i^2 = -1$.

Subset

A set $A$ is a subset of $B$ (written $A \subseteq B$) if every element of $A$ is also an element of $B$.

Proper Subset

A set $A$ such that $A \subseteq B$ but $A \neq B$, symbolically denoted as $A \subsetneq B$.

Set Intersection

The set operation producing $A \cap B$, containing all elements belonging to both set $A$ and set $B$ ($(x \in A) \land (x \in B)$).

Disjoint Sets

Two sets $A$ and $B$ that share no common elements, meaning $A \cap B = \emptyset$.

Set Union

The set operation producing $A \cup B$, containing all items that are elements of $A$, of $B$, or of both ($(x \in A) \lor (x \in B)$).

Set Complement

The set $A^c$, with respect to a universal set $U \supseteq A$, consisting of all elements in $U$ that are not in $A$ ($A^c = U \setminus A$).

Set Difference

The set $A \setminus B$ consisting of all elements that belong to set $A$ but do not belong to set $B$, defined as $(x \in A) \land (\neg(x \in B))$.

Cardinality

The total number of elements contained in a finite set $A$, denoted symbolically by $|A|$.

Partition of a Set

A decomposition of a set $A$ into a collection of pairwise disjoint subsets such that every element of $A$ belongs to exactly one subset.

Principle of Inclusion and Exclusion

A counting identity for finite sets stating that $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$.

Cartesian Product

The set $A \times B = \{(a, b) \mid (a \in A) \land (b \in B)\}$ of all ordered pairs whose first element belongs to $A$ and second belongs to $B$, with size $|A \times B| = |A| \cdot |B|$ for finite sets.

Binary Relation

A subset $R \subseteq A \times B$ from domain $A$ to codomain $B$, where $(x, y) \in R$ indicates that $x$ and $y$ are related.

Reflexive Relation

A relation $R$ on a set $A$ where every element is related to itself, meaning $(x, x) \in R$ for all $x \in A$.

Irreflexive Relation

A relation $R$ on a set $A$ where no element is related to itself, meaning $(x, x) \notin R$ for all $x \in A$.

Symmetric Relation

A relation $R$ on a set $A$ such that for all $(x, y) \in A \times A$, $(x, y) \in R$ implies that $(y, x) \in R$.

Anti-symmetric Relation

A relation $R$ on a set $A$ where whenever $(x, y) \in R$ and $(y, x) \in R$, it necessarily follows that $x = y$.

Transitive Relation

A relation $R$ on a set $A$ where if $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$ must also hold.

Equivalence Relation

A relation on a set that satisfies reflexivity, symmetry, and transitivity simultaneously.

Equivalence Class

The set $E_x := \{y \in A \mid (y, x) \in R\}$ of all elements related to element $x$ under an equivalence relation $R$.

$n$-ary Relation

A subset $R \subseteq A_1 \times A_2 \times \dots \times A_n$ composed of ordered $n$-tuples, where each set $A_i$ specifies the allowed domain for coordinate $i$.

Relational Model

The database framework proposed by E. F. Codd in 1970 that applies set theory by modeling database tables as mathematical relations, rows as ordered tuples, and columns as attributes.

Function

A relation $f$ from set $A$ to set $B$ satisfying existence (every $a \in A$ maps to some $b \in B$) and uniqueness (if $(a, b_1) \in f$ and $(a, b_2) \in f$, then $b_1 = b_2$).

Identity Function

The function on a set $A$ denoted by $\text{id}: A \to A : a \mapsto a$, which maps every element to itself.

Linear Function

A function $f: \mathbb{R} \to \mathbb{R}$ of the form $f(x) = mx + b$, whose graph is a straight line with slope $m$ and intercept $b$.

Quadratic Function

A function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = ax^2 + bx + c$ with $a \neq 0$, whose graph forms a parabola (cup-shaped when $a > 0$ and cap-shaped when $a < 0$).

Extremum Point (Quadratic)

The single minimum ($a > 0$) or maximum ($a < 0$) point of $f(x) = ax^2 + bx + c$, occurring at coordinate $x = -\frac{b}{2a}$ with value $f(-\frac{b}{2a}) = \frac{4ac - b^2}{4a}$.

Discriminant

The expression $\Delta = b^2 - 4ac$ from the quadratic formula, indicating two distinct real roots if $\Delta > 0$, one repeated real root if $\Delta = 0$, and no real roots if $\Delta < 0$.

Roots of a Function

The $x$-values (zeroes) where $f(x) = 0$, corresponding geometrically to the points where the function's graph intersects the $x$-axis.

Function Composition

The process of combining two functions $f: A \to B$ and $g: B \to C$ to form a new function $g \circ f: A \to C$, defined by $(g \circ f)(x) = g(f(x))$.

Injective Function

A function $f: A \to B$ where for every $b \in B$ there is at most one $a \in A$ satisfying $b = f(a)$.

Surjective Function

A function $f: A \to B$ where for every $b \in B$ there is at least one $a \in A$ such that $b = f(a)$, meaning the range equals the entire codomain $B$.

Bijective Function

A function that is simultaneously injective and surjective; a function is invertible if and only if it is bijective.

Inverse Function

A function $f^{-1}: B \to A$ associated with an invertible function $f: A \to B$ such that $(f^{-1} \circ f)(a) = a$ and $(f \circ f^{-1})(b) = b$.

Product Demand ($Q_d$)

The quantity of a good that buyers wish to purchase, modeled linearly as $Q_d(p) = mp + b$ where the slope is negative ($m < 0$) and intercept is positive ($b > 0$).

Product Supply ($Q_s$)

The quantity of a good available for sale, modeled linearly as $Q_s(p) = rp + d$ where the slope is positive ($r > 0$) and intercept is negative ($d < 0$).

Market Equilibrium

The economic state where product supply equals product demand ($Q_s(p) = Q_d(p)$), clearing the market so all produced goods are sold.

Fixed Costs ($f$)

Production expenses incurred by a company that do not vary with the amount produced, such as facility maintenance ($f > 0$).

Variable Costs ($v$)

Production expenses that scale proportionally with each additional unit of product manufactured, such as raw material costs ($v > 0$).

Production Cost Function ($C$)

The total cost to produce a given supply $Q_s$, modeled as $C(Q_s) = v Q_s + f$, or evaluated as a function of price via $C(p) = C(Q_s(p))$.

Revenue ($R$)

The total sales earnings of a company given by $R = p \cdot Q_d$, which models as a concave downward quadratic function of price $R(p) = mp^2 + bp$ when demand is linear.

Profit ($P$)

The financial gain of a firm calculated as total revenue minus total production costs ($P = R - C$).

Break-Even Point

The operating state where total profit is zero ($P = 0$), meaning total revenue precisely offsets total production costs.