Set Notation, Geometry, Number Systems, and Algebraic Properties

Vocabulary flashcards generated from lecture notes on set notation, geometric definitions, number classifications, and algebraic properties of equality.

Set

A list of elements (e.g., $A = \{a, b, c, d, \dots\}$).

Element ($a \in X$)

An item on a list, indicating that $a$ is an element of set $X$.

Subset ($X \subset Y$)

A relationship where every element of set $X$ is also an element of set $Y$.

Union ($X \cup Y$)

The set containing all the elements of $X$ and all the elements of $Y$.

Intersection ($X \cap Y$)

The set containing all the elements that are in both $X$ and $Y$ (their commonalities).

Empty Set

The set containing no elements, representing no solution (denoted as $\emptyset$ or $0$).

Point

A singular location, named using a single capital letter (e.g., $L$).

Line

A series of points that go infinitely in either direction, named using 2 letters with arrows above them (e.g., $\overleftrightarrow{AB}$).

Line Segment

A series of points that truncate with endpoints, named using 2 letters with a bar and no arrows above them (e.g., $\overline{AB}$).

Ray

A series of points that go infinitely in one direction and has an endpoint in the other, named with an arrow facing the correct way relative to the lettering (e.g., $\overrightarrow{AB}$ or $\overleftarrow{BA}$).

Plane

A two-dimensional, flat surface that extends without end, named by a script letter or 3 points (e.g., $m$ or $ABC$).

Opposite Rays

Two rays that share the same endpoint and extend in opposite directions infinitely, where the three points must be collinear (e.g., $\overrightarrow{AB}$ and $\overrightarrow{BC}$).

Collinear

Points that lie on the same line.

Coplanar

Points or lines that lie on the same plane.

Natural Numbers ($N$)

Any positive whole number, represented in set notation as $\{1, 2, 3, \dots\}$.

Integers ($Z$)

All whole numbers (positive, negative, and zero), represented in set notation as $\{\dots, -1, 0, 1, \dots\}$.

Rational Numbers ($Q$)

Any number that can be written as a fraction (e.g., $\{\frac{3}{4}, \frac{1}{2}, 2, \dots\}$).

Irrational Numbers

Any number that cannot be written as a fraction (e.g., $\{\sqrt{16}, \sqrt{7}, \pi, \sqrt{2}, \sqrt{5}, \sqrt{3}\text{...}\}$).

Real Numbers ($R$)

Any number that is not imaginary; formed by the union of rational and irrational numbers.

Commutative Property of Addition

State that changing the order of addends does not change the sum: $a + b = b + a$.

Commutative Property of Multiplication

States that changing the order of factors does not change the product: $ab = ba$.

Associative Property of Addition

States that changing the grouping of addends does not change the sum: $(a + b) + c = a + (b + c)$.

Associative Property of Multiplication

States that changing the grouping of factors does not change the product: $(ab)c = a(bc)$.

Distributive Property

Multiplying a sum by a number gives the same result as multiplying each addend individually: $a(b + c) = ab + ac$.

Addition Property of Equality

If $a = b$, then $a + c = b + c$.

Subtraction Property of Equality

If $a = b$, then $a - c = b - c$.

Multiplication Property of Equality

If $a = b$, then $ac = bc$.

Division Property of Equality

If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

Transitive Property of Equality

If $a = b$ and $b = c$, then $a = c$.