Mathematics in the Modern World - Vocabulary Flashcards

Vocabulary practice flashcards covering core concepts of mathematics in the modern world, including the Fibonacci sequence, propositions, logical connectives, truth tables, equivalences, quantifiers, set theory, and counting principles.

Leonardo Bonacci (Fibonacci)

A medieval mathematician and businessman born in 1170 in Pisa, Italy, popularly known as Fibonacci (short for filius Bonacci, meaning 'son of Bonacci') and also known as Leonardo Pisano Bigollo.

Liber Abaci

A famous book written by Fibonacci in 1202, which means 'The Book of Calculation'.

Fibonacci Sequence

A recursive sequence obeying the rule that each term is calculated by summing the preceding two terms, given by $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, \text{...}$

Golden Ratio

A special mathematical ratio associated with the Fibonacci sequence, given as $1 : (1.61803...)$.

Proposition

A declarative statement which is true or false, but not both.

Truth Value

The attribute assigned to a proposition depending on its truthfulness or falsehood, which in classical logic has two possible values: true or false.

Propositional Logic

The study of how simple propositions come together using logical connectives to make more complicated propositions.

Logical Operators

Connectives used to form compound propositional molecules from simple atomic propositions, including negation, conjunction, disjunction, conditional, and biconditional.

Negation ($\neg p$)

A logical operator interpreted intuitively as being true when $p$ is false, and false when $p$ is true ('not').

Conjunction ($p \wedge q$)

A compound proposition that is true only when both simple propositions $p$ and $q$ are true ('and', 'but').

Disjunction ($p \vee q$)

A compound proposition that is true when at least one of the two propositions is true, corresponding to an inclusive 'or'.

Conditional Statement ($p \rightarrow q$)

A compound proposition ('if $p$, then $q$') that is false only when the hypothesis $p$ is true while the conclusion $q$ is false.

Hypothesis

The antecedent or premise proposition $p$ in a conditional statement $p \rightarrow q$.

Conclusion

The consequence proposition $q$ in a conditional statement $p \rightarrow q$.

Biconditional Statement ($p \leftrightarrow q$)

A compound proposition ('$p$ if and only if $q$') that is true when $p$ and $q$ have the same truth values, and false otherwise.

Truth Table

A mathematical table showing how the truth or falsity of a compound proposition varies with that of its components.

Logical Equivalence

The property of two compound propositions $P$ and $Q$ (written $P \equiv Q$ or $P \Leftrightarrow Q$) having identical truth tables for all interpretations of their component variables.

Converse

A conditional statement formed by interchanging the hypothesis and conclusion of the original conditional statement, represented as $q \rightarrow p$ for $p \rightarrow q$.

Inverse

A conditional statement formed by negating the hypothesis and negating the conclusion of the original statement, represented as $\neg p \rightarrow \neg q$ for $p \rightarrow q$.

Contrapositive

A conditional statement formed by negating both the hypothesis and conclusion and interchanging their order, represented as $\neg q \rightarrow \neg p$ for $p \rightarrow q$.

Tautology

A compound proposition whose truth table contains only true values for every case.

Contradiction

A compound proposition whose truth table contains only false values for every case.

Contingency

A compound proposition whose truth table contains at least one row where it is true and at least one row where it is false.

Predicate

A propositional function in logic expressed in a way that assumes a value of true or false once values are assigned to its variables or quantified.

Universe of Discourse

The domain or set of all possible values from which the variable of a propositional function can take values.

Universal Quantification ($$\forall x P(x)$Static)

The proposition stating that $P(x)$ is true for all values of $x$ in the universe of discourse.

Existential Quantification ($$\exists x P(x)$Static)

The proposition stating that there exists an element $x$ in the universe of discourse such that $P(x)$ is true.

Set

A collection of well-defined objects, whose members are called elements.

Roster Method

A way of describing a set by listing all of its members (or using ellipses when a pattern is obvious) between braces.

Set-Builder Notation

A method of characterizing elements in a set by stating the property or properties they must satisfy to be members.

Cardinality

The number of elements in a set $A$, denoted as $|A|$ or $n(A)$.

Finite Set

A set which either has no elements or has elements that can all be listed down (countable).

Infinite Set

A set whose elements cannot be listed because they are unlimited in quantity.

Universal Set ($U$)

A set containing all of the elements relevant to a given discussion.

Subset ($A \subseteq B$)

A relationship where every element of Set $A$ is also contained in Set $B$.

Proper Subset ($A \subset B$)

A relationship where $A \subseteq B$ but $A \neq B$.

Null Set

A set with no elements, denoted in symbols by $\emptyset$ or $\{\}$.

Venn Diagram

A visual tool used for illustrating set-theoretic relationships using a rectangle for the universal set and ovals for its subsets.

Union ($A \cup B$)

The set operation producing a set which contains all the elements present in both sets $A$ and $B$.

Intersection ($A \cap B$)

The set operation producing a set containing only the elements that are common to both sets $A$ and $B$.

Set Difference ($A - B$)

The set operation producing a set whose elements are found in the first set ($A$) but not in the second set ($B$).

Set Complement ($\bar{A}$)

The set operation producing a set whose elements are in the universal set ($U$) but not in the given set ($A$).

Principle of Inclusion-Exclusion

A counting technique that generalizes the method of obtaining the number of elements in the union of finite sets, expressed for two sets as $|A \cup B| = |A| + |B| - |A \cap B|$.