Mathematics in the Modern World: Final Exam Reviewer

Vocabulary practice flashcards covering statistics (standard normal distribution, z-scores, skewness, kurtosis), graph theory (equivalent graphs, Euler and Hamiltonian circuits, algorithms), and mathematical logic (compound statements, truth tables, quantified statements, and De Morgan's laws).

Standard Normal Distribution (Empirical Rule)

A rule for normal distributions stating that approximately $68\%$ of observations fall within $1$ standard deviation of the mean ($\mu \pm 1\sigma$), approximately $95\%$ fall within $2$ standard deviations ($\mu \pm 2\sigma$), and approximately $99.7\%$ fall within $3$ standard deviations ($\mu \pm 3\sigma$).

z-score

A standardized measure that indicates the number of standard deviations a data value $x$ falls above or below the mean $\mu$, given by the formula $z = \frac{x - \mu}{\sigma}$.

Actual Score Formula

The formula used to calculate a raw score $x$ when the $z$-score, population standard deviation $\sigma$, and mean $\mu$ are known: $x = (z \times \sigma) + \mu$.

Positive Skew

A distribution that is skewed to the right, where the tail extends toward higher values, most observations are clustered at the lower end, and the order of central tendency measures is $\text{Mode} < \text{Median} < \text{Mean}$.

Negative Skew

A distribution that is skewed to the left, where the longer tail extends toward lower values, most observations are concentrated at the higher end, and the order of central tendency measures is $\text{Mean} < \text{Median} < \text{Mode}$.

Symmetrical Distribution

A distribution in which the data is evenly distributed around the center, resulting in equal values for central tendency: $\text{Mean} = \text{Median} = \text{Mode}$.

Platykurtic

A description of a distribution curve whose central peak is lower and broader, and whose tails are shorter and thinner relative to a normal distribution.

Leptokurtic

A description of a distribution curve whose central peak is higher and sharper, and whose tails are longer and fatter relative to a normal distribution.

Kurtosis

The statistical measure of the peakedness and flatness of the center and the weight of the tails of a probability distribution curve, classified into leptokurtic, mesokurtic, and platykurtic.

Equivalent Graphs

Two graphs that have the same adjacency relationships, meaning the exact same pairs of vertices are connected by edges.

Degree of a Vertex

The number of edges that meet or are incident at a specific vertex in a graph.

Eulerian Graph Theorem

A theorem stating that a connected graph is Eulerian (contains an Euler circuit) if and only if every vertex in the graph has an even degree.

Euler Path Theorem

A theorem stating that a connected graph contains an Euler path if and only if the graph has exactly $2$ vertices of odd degree while all other vertices have an even degree, requiring the path to start at one odd-degree vertex and end at the other.

Hamiltonian Circuit

A closed path in a graph that visits every vertex exactly once and returns to the initial starting vertex.

Weighted Graph

A graph in which a real number value, known as a weight (such as distances in miles or lengths of time), is assigned to every edge.

The Edge-Picking Algorithm

An algorithm for weighted graphs where one begins by marking the edge with the smallest weight in the graph, choosing any one arbitrarily if multiple edges tie for the lowest weight.

The Greedy Algorithm

An algorithm where one selects a starting vertex and repeatedly travels along the connected edge having the smallest weight, choosing any edge arbitrarily if there is a tie.

Statement (Proposition)

A declarative sentence that is either true or false, but not both true and false at the same time.

Simple Statement

A declarative statement that does not contain any logical connective.

Negation

A logical compound statement using the connective "not", represented symbolically as $\sim p$, which reverses the truth value of statement $p$.

Conjunction

A compound statement formed using the connective "and", written symbolically as $p \wedge q$, which is true only when both component statements $p$ and $q$ are true.

Disjunction

A compound statement formed using the connective "or", written symbolically as $p \vee q$, which is false only when both component statements $p$ and $q$ are false.

Conditional Statement

A compound statement formed using the connective "If… then", denoted symbolically as $p \rightarrow q$, which is false only when the antecedent $p$ is true and the consequent $q$ is false.

Biconditional Statement

A compound statement formed using the connective "if and only if", denoted symbolically as $p \leftrightarrow q$, which is true when both component statements share the same truth value.

Negation of "All X are Y"

The logically equivalent quantified statement "Some X are not Y".

Negation of "No X are Y"

The logically equivalent quantified statement "Some X are Y".

Tautology

A compound statement that is always true for every possible combination of truth values of its constituent simple statements.

Self-Contradiction

A compound statement that is always false for every possible combination of truth values of its constituent simple statements.

Contingency

A compound statement whose truth value is neither always true nor always false; that is, a statement that is neither a tautology nor a self-contradiction.

De Morgan’s Laws for Statements

Pair of equivalence laws stating that the negation of a disjunction is equivalent to a conjunction of negations, $\sim(p \vee q) \equiv \, \sim p \wedge \sim q$, and the negation of a conjunction is equivalent to a disjunction of negations, $\sim(p \wedge q) \equiv \, \sim p \vee \sim q$.