Mathematical Language and Symbols Vocabulary Flashcards

Comprehensive vocabulary flashcards covering concepts, definitions, and terminology from Chapter II: Mathematical Language and Symbols.

Precision

The quality or condition of being exact and accurate in mathematical language, allowing one to make very fine distinctions.

Conciseness

The characteristic of using the most appropriate and minimal number of effective words to make one's point understood.

Powerful (Language Characteristic)

The ability of mathematical language to express complex thoughts with relative ease.

Grammar of Mathematics

The structural rules governing the use of symbols representing mathematical objects.

Expression

The mathematical analogue of a noun; a name given to a mathematical object of interest such as a number, set, matrix, or ordered pair.

Mathematical Sentence

The mathematical analogue of an English sentence; a correct assignment of mathematical symbols that states a complete thought and possesses a truth value.

Mathematical Conventions

Particular symbols, facts, names, and notations used by mathematicians, engineers, scientists, and other users of mathematics in their writings and work.

Set

A well-defined collection of distinct objects.

Cardinality of a Set

Refers to the number of elements contained in a set $A$, denoted by $n(A)$.

Roster Method

A method of writing a set by listing its elements enclosed in curly brackets.

Rule Method

A method of writing a set by describing its elements.

Universal Set

The large fixed set, denoted by $U$, assumed to contain all sets under investigation.

Empty Set

A set containing no elements, also called a null set, denoted by $\{\}$ or $\emptyset$.

Finite Set

A set consisting of elements in which the number of elements is countable.

Infinite Set

A set consisting of elements in which the number of elements is not countable or indefinite.

Subset

A set taken from another set, denoted by $B \subseteq A$ if every element of set $B$ is also in set $A$.

Proper Subset

A subset that is not identical to the original set and contains fewer elements, denoted using the symbol $\subset$.

Improper Subset

A subset whose elements are identical to the original set, as well as the empty set.

Equal Sets

Two or more sets that have the exact same elements regardless of order.

Equivalent Sets

Two or more sets that have the same number of elements ($n(A) = n(B)$).

Joint Sets

Two or more sets that have at least one common element.

Disjoint Sets

Two or more sets that do not have any common elements.

Union of Sets

The operation on two sets $A$ and $B$, denoted $A \cup B$, containing all elements contained in either set, both sets, or all sets.

Intersection of Sets

The operation on two sets $A$ and $B$, denoted $A \cap B$, containing only the elements that are present in both sets.

Difference of Sets

The operation on two sets $A$ and $B$, denoted $A - B$, containing elements found in $A$ but not in $B$.

Complement of a Set

The set containing everything in the universal set $U$ that is not in set $A$, denoted $A'$, $A^c$, or $\sim A$.

Venn Diagram

A pictorial representation of sets using enclosed areas in the plane, where the universal set $U$ is represented by a rectangle and other sets by circles inside.

Relation

A correspondence between two things or quantities represented as a set of ordered pairs $(x, y)$.

Domain

The set of all input values or $x$-values of a relation or function.

Range

The set of all output values or $y$-values as $x$ varies over the domain in a relation or function.

Cartesian Product

The set $A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}$, consisting of all ordered pairs formed from sets $A$ and $B$.

Function

A rule from domain $D$ to set $R$ that associates or assigns to each element in $D$ a single unique element in $R$.

Composite Function

A combination of two functions $f$ and $g$, denoted $(f \circ g)(x) = f(g(x))$, where the domain consists of all $x$ in the domain of $g$ for which $g(x)$ is in the domain of $f$.

Logic

The study of correct thinking and reasoning using principles and methods to distinguish valid arguments from invalid ones.

Statement (Proposition)

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

Simple Statement

A statement that conveys a single idea.

Compound Statement

A statement that conveys two or more ideas by connecting simple statements with logical connectives.

Logical Connective

A word or symbol that joins two sentences to produce a new one.

Negation

A statement formed by taking the opposite truth value of proposition $p$, denoted by $\sim p$.

Conjunction

A compound statement formed using 'and', denoted $p \land q$, which is true only when both component statements are true.

Disjunction

A compound statement formed using 'or', denoted $p \lor q$, which is true if at least one of the component statements is true.

Conditional Statement (Implication)

A compound statement formed using 'If… then', denoted $p \rightarrow q$, which is false only when $p$ is true and $q$ is false.

Biconditional Statement

A compound statement formed using 'if and only if', denoted $p \leftrightarrow q$, which is true when both component statements have the same truth value.