Set Theory and Symbolic Logic Fundamentals

Comprehensive vocabulary flashcards covering the basic principles of set theory, set operations, symbolic logic, compound statements, truth values, and quantifiers from Aufmann's Mathematics text.

Set

Any group or collection of objects.

Elements (or Members)

The objects that belong in a set.

Roster Method

A method of designating a set by listing its elements inside a pair of braces, $\left\{ \right\}$, with commas used to separate the elements.

Ellipsis ($\dots$)

Three dots indicating that the elements of a set continue in a manner suggested by the elements that are listed.

Natural Numbers (Counting Numbers)

The basic set of numbers denoted by $N = \left\{ 1, 2, 3, 4, 5, \dots \right\}$, also referred to as the positive integers.

Whole Numbers

The basic set of numbers denoted by $W = \left\{ 0, 1, 2, 3, 4, 5, \dots \right\}$.

Integers

The set of numbers denoted by $I = \left\{ \dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots \right\}$.

Rational Numbers

The set of numbers denoted by $Q$, consisting of all terminating or repeating decimals, which can also be written in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

Irrational Numbers

The set of numbers consisting of all nonterminating, nonrepeating decimals.

Real Numbers

The set of numbers denoted by $R$, formed by combining the rational numbers and the irrational numbers.

Well-Defined Set

A set for which it is possible to determine whether any given item is an element of the set.

Cardinal Number (Cardinality)

The number of elements in a finite set. For a finite set $A$, it is denoted by $n(A)$.

Equal Sets

Two sets $A$ and $B$ (denoted $A = B$) that have exactly the same elements.

Universal Set

The set of all elements that are being considered in a particular discussion or problem, denoted by the letter $U$.

Complement of a Set

The set of all elements of the universal set $U$ that are not elements of set $A$, denoted by $A'$.

Empty Set (or Null Set)

The set that contains no elements, denoted by $\emptyset$ or $\left\{ \right\}$.

Subset

Set $A$ is a subset of set $B$, denoted by $A \subseteq B$, if and only if every element of $A$ is also an element of $B$.

Proper Subset

Set $A$ is a proper subset of set $B$, denoted by $A \subset B$, if every element of $A$ is an element of $B$, and $A \neq B$.

Number of Subsets Formula

A mathematical formula stating that a set with $n$ elements has $2^n$ subsets and $2^n - 1$ proper subsets.

Venn Diagram

A diagram developed by John Venn used to illustrate sets and relationships between sets, where the universal set is represented by a rectangular region and subsets are represented by oval or circular regions inside.

Intersection of Sets

The set of elements common to both set $A$ and set $B$, denoted by $A \cap B$.

Union of Sets

The set that contains all the elements that belong to set $A$ or to set $B$ or to both, denoted by $A \cup B$.

Disjoint Sets

Two sets whose intersection is the empty set ($A \cap B = \emptyset$).

Counterexample

A specific case or example used to prove that a general mathematical statement or set expression equality is false.

De Morgan's Laws (Sets)

The two set identities stating that $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

Statement (Logic)

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

Open Statement

A declarative sentence containing a variable (such as $x + 1 = 5$) that is true or false depending on the value assigned to the variable.

Compound Statement

A statement formed by connecting two or more component statements with words and phrases such as and, or, not, if … then, and if and only if.

Connective

A word or phrase (such as and, or, not, if … then, or if and only if) used to connect component statements.

Truth Value

The classification of a statement as true ($\text{T}$) if it is true, or false ($\text{F}$) if it is false.

Truth Table

A table that shows the truth values of a statement for all possible truth values of its component statements.

Negation

The logical denial of a statement $p$, denoted by $\sim p$, which has the opposite truth value of $p$.

Conjunction

A compound statement formed using the connective and, denoted symbolically by $p \land q$, which is true if and only if both $p$ and $q$ are true.

Disjunction

A compound statement formed using the connective or, denoted symbolically by $p \lor q$, which is true if $p$ is true, if $q$ is true, or if both $p$ and $q$ are true.

Existential Quantifiers

The word some and the phrases there exists and at least one, used as prefixes to assert the existence of something.

Universal Quantifiers

The words none, no, all, and every, used to assert that every element of a given set satisfies a condition or to deny the existence of something.