Patterns in Nature, Mathematical Language, and Logic

Flashcard deck covering patterns in nature, the etymology of mathematics, the Fibonacci sequence, characteristics of mathematical language, formal logic, propositions, truth tables, conditional statements, and quantifiers.

What is the etymology and original meaning of the word mathematics?

The word mathematics comes from the Greek word Mathema, which means "that which is learnt" in Ancient Greek and "lesson" in Modern Greek. It is derived from Manthano (equivalent to mathaino in modern Greek), which means "to learn".

How does Encyclopedia Britannica define mathematics?

Encyclopedia Britannica defines mathematics as "the science of structure, order, and relations that has evolved from elemental practices of counting, measuring, and describing the shapes and characteristics of objects."

According to Villa (2016), what is the definition of the Fibonacci sequence?

The Fibonacci sequence is an infinite sequence of natural numbers where the first value is 0, the next is 1, and from there, each succeeding value is obtained by adding the previous two.

What is the recursive formula for determining the $n\text{th}$ Fibonacci number for $n \ge 2$?

Given $F_0 = 0$ and $F_1 = 1$, the recursive formula is $F_n = F_{n-2} + F_{n-1}$.

What is Binet's explicit formula for calculating the $n\text{th}$ Fibonacci number $F_n$?

$$F_n = \frac{1}{\sqrt{5}} \left[ \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right]$$

What constant numerical value does the ratio of two consecutive Fibonacci numbers $\frac{a}{b}$ (where $a > b$) approach?

It approaches the Golden Ratio $\Phi$ (Phi), which is approximately $1.61803399...$

What is the approximate value of the Golden Angle $x$?

$$x = 137.507764^\circ... \approx 137.5^\circ$$

What are the three primary characteristics of mathematical language?

Precise (able to make very fine distinctions), Concise (able to say things briefly), and Powerful (able to express complex thoughts with relative ease).

How does a mathematical expression differ from a mathematical sentence?

A mathematical expression is analogous to a noun (the name given to a mathematical object of interest), whereas a mathematical sentence states a complete mathematical thought and possesses a truth value.

What is a proposition in formal logic?

A proposition is a declarative sentence that expresses a complete thought and is either true or false, but not both.

How is an even number formally defined?

An even number is an integer with the form $n = 2k$, where $k \in \mathbb{Z}$.

What is a logical connective?

A logical connective is the mathematical equivalent of a conjunction—a word or symbol that joins two sentences to produce a new one.

What is the symbol, reading, and truth condition for a logical conjunction of $p$ and $q$?

Symbol: $p \land q$ (read as "$p$ and $q$"). It is true if and only if both $p$ and $q$ are true.

Under what condition is the disjunction $p \lor q$ true?

A disjunction $p \lor q$ is true if and only if $p$ is true, $q$ is true, or both are true.

Under what specific circumstance is the conditional implication $p \rightarrow q$ false?

An implication $p \rightarrow q$ is false only when $p$ is true and $q$ is false. It is true under all other circumstances.

What does it mean for two propositions to be logically equivalent ($p \equiv q$)?

Two propositions are logically equivalent if and only if they have the same truth value.

Given a conditional proposition $p \rightarrow q$, what are its converse, inverse, and contrapositive?

Converse: $q \rightarrow p$; Inverse: $\neg p \rightarrow \neg q$; Contrapositive: $\neg q \rightarrow \neg p$.

Which statement form is logically equivalent to the conditional implication $p \rightarrow q$?

The contrapositive, $\neg q \rightarrow \neg p$.

What is the Universal Quantifier, its symbol, and its meaning?

Symbol: $\forall$; Meaning: "For all" or "For every".

What is the Existential Quantifier, its symbol, and its meaning?

Symbol: $\exists$; Meaning: "There exists" or "For some".

What is the logical equivalence for the negation of a universally quantified statement, $\neg \forall x P(x)$?

$$\neg \forall x P(x) \equiv \exists x \neg P(x)$$

What is the logical equivalence for the negation of an existentially quantified statement, $\neg \exists x P(x)$?

$$\neg \exists x P(x) \equiv \forall x \neg P(x)$$