Problem Solving and Reasoning - Modules 1 to 3

Vocabulary flashcards covering the nature of mathematics, mathematical language and symbols, logic, set theory, functions, reasoning, and financial mathematics from Modules 1 through 3.

Mathematics

The study of patterns and relationships in numbers, shapes, structures, and space, serving as a language and human endeavor to describe, explain, and understand the world.

Patterns

Regular arrangements or repeated relationships found in numbers, shapes, structures, and natural phenomena.

Symmetry

A balanced and proportionate similarity found in two halves of an object, pattern, or shape.

Translational Symmetry

A type of symmetry where a shape, object, or pattern is moved or slid a specific distance in a straight direction and still looks identical, changing location while preserving size, shape, and angles.

Rotational Symmetry

A type of symmetry where an object looks similar after being rotated around a point.

Reflection Symmetry

A type of symmetry where one part of an object resembles the mirror image of another part.

Glide Symmetry

Also called glide reflection symmetry; a type of symmetry where an object or pattern looks identical after being flipped across an axis and slid forward a specific distance.

Line of Symmetry

A line that divides a figure into two parts that are mirror images of each other, showing how the figure is balanced.

Sequence

An ordered list of numbers that follows a specific rule or pattern, where each individual number is called a term.

Fibonacci Sequence

A sequence associated with Leonardo of Pisa where each term is obtained by adding the two preceding terms, governed by the formula $F_n = F_{n-1} + F_{n-2}$.

Golden Ratio ($\phi$)

An irrational constant approximately equal to $1.618$, defined when whole : larger part = larger part : smaller part, which the ratio of consecutive Fibonacci numbers approaches as they become larger.

Golden Spiral

A spiral pattern connected to the Golden Ratio, observed in natural patterns such as shell spirals, hurricanes, galaxies, and tree branching.

Mathematical Expression

A combination of mathematical symbols that represents an object or quantity but does not make a claim (for example, $3t + 5$).

Mathematical Statement

A mathematical phrase that makes a claim that can be evaluated as true or false for particular values (for example, $3t + 6 = 9$).

Proposition

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

Quantifiers

Symbols used to specify quantity in logical statements, such as Universal Quantification $\forall$ (for all / every) and Existential Quantification $\exists$ (there exists).

PEMDAS

The standard order of operations used to simplify mathematical expressions: Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction.

Negation ($\neg p$)

A logical connective representing NOT $p$, which reverses the truth value of a proposition.

Conjunction ($p \land q$)

A logical connective representing $p$ AND $q$, which is true only when both propositions $p$ and $q$ are true.

Disjunction ($p \lor q$)

A logical connective representing $p$ OR $q$, which is false only when both propositions $p$ and $q$ are false.

Conditional ($p \rightarrow q$)

A logical connective representing IF $p$, THEN $q$, which is false only when $p$ is true and $q$ is false.

Biconditional ($p \leftrightarrow q$)

A logical connective representing $p$ IF AND ONLY IF $q$, which is true when both propositions $p$ and $q$ have the same truth value.

Set

A well-defined collection of objects.

Element

An individual object, number, or item that belongs to a set, indicated by the symbol $\in$.

Cardinality

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

Empty Set

Also called a null set; a set that contains no elements, denoted by $\emptyset$ or $\{\}$.

Finite Set

A set containing a limited and countable number of elements.

Infinite Set

A set containing an unlimited number of elements that continues indefinitely.

Union ($A \cup B$)

A set operation combining all elements that belong to set $A$, set $B$, or both.

Intersection ($A \cap B$)

A set operation containing only the elements common to both set $A$ and set $B$.

Difference ($A - B$)

A set operation consisting of all elements that belong to set $A$ but do not belong to set $B$.

Complement ($A'$)

A set operation consisting of all elements in the universal set that are not in set $A$.

Roster Method

A way of describing a set by explicitly listing all of its individual elements within braces.

Set-Builder Form

A way of describing a set using a rule or condition that its elements must satisfy (for example, \x \mid 1 \le x \le 5\}).

Relation

An association between elements of two sets, commonly represented as a set of ordered pairs.

Function

A special type of relation that assigns exactly one output to every input.

Domain

The set of all possible input values for a relation or function.

Range

The set of all resulting output values produced by a relation or function.

Reasoning

The process of using facts, patterns, and logical thinking to arrive at conclusions and justify mathematical statements.

Inductive Reasoning

A form of reasoning that involves observing specific examples or patterns to arrive at a general conclusion (Specific $\rightarrow$ General).

Deductive Reasoning

A form of reasoning that starts with accepted facts or premises to arrive at a logically certain conclusion (General $\rightarrow$ Specific).

Polya's Four Steps

A framework for problem solving consisting of four steps: 1. Understand the Problem, 2. Devise a Plan, 3. Carry out a plan, and 4. Look Back.

Simple Interest

Interest calculated only on the original principal amount, calculated using $I = Prt$ and total amount $A = P + I$.

Compound Interest

Interest calculated on both the initial principal and the accumulated interest from previous periods, calculated using $FV = PV(1 + r)^n$.

Tautology

A compound statement in logic that is true for all possible truth values of its component statements.

Contradiction

A compound statement in logic that is false for all possible truth values of its component statements.

Contingency

A compound statement in logic that can be either true or false depending on the truth values of its component statements.