Sets, Numbers, and Algebraic Expressions Flashcards

Flashcards covering set notation, types of real numbers, inequality symbols, interval notation, and algebraic properties based on the lecture notes.

Set

A collection of things, called elements, usually indicated by brackets ${}$.

Natural numbers

The set represented by ${1, 2, 3, \dots }$.

Whole numbers ($W$)

The set represented by ${0, 1, 2, 3, \dots }$.

Integers

The set represented by ${ \dots , -3, -2, -1, 0, 1, 2, 3 \dots }$.

$$\in$$

A symbol meaning "is an element of" (belongs to).

$$\notin$$

A symbol meaning "is not an element of."

$$\subset$$

A symbol meaning "is a proper subset of," which indicates a set is contained in another but not equal to it.

Rational numbers ($Q$)

All numbers that can be written as fractions of two integers or as terminating or repeating decimal numbers.

Real numbers ($R$)

All numbers that may be represented as a length, in the forward ($+$) or backward ($-$) direction on the number line.

Irrational numbers

Real numbers that are not rational and cannot be written as fractions of integers or repeating or terminating decimals, such as $\sqrt{2}$ or $\pi$.

$$\leq$$

An inequality symbol meaning "is less than or equal to," "is no more than," or "is at most."

$$\geq$$

An inequality symbol meaning "is greater than or equal to," "is at least," or "is no less than."

Least common denominator (LCD)

The smallest common multiple of two denominators, used to compare or add/subtract fractions with different denominators.

Interval notation ($a, b$)

Represents all real numbers between $a$ and $b$, but not including $a$ or $b$.

Interval notation $[a, b]$

Represents all real numbers between $a$ and $b$, including both $a$ and $b$.

Union ($\cup$)

The collection of all elements that are in set $A$ or in set $B$ or in both.

Intersection ($\cap$)

The collection of elements that are both in set $A$ and in set $B$.

Absolute value

A value defined as $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$, representing the distance from zero on a number line.

PEMDAS

An acronym for the order of operations: Parentheses (innermost first), Exponents, Multiply and Divide (left to right), and Add and Subtract (left to right).

Multiplicative inverse

A property where for any $a \neq 0$, $a \times (\frac{1}{a}) = 1$.

Distributive Law

The algebraic property stated as $a(b + c) = ab + ac$.

Like terms

Terms that have the same variables, each raised to the same power.