Mathematics Vocabulary Review (Algebra, Inequalities, and Set Theory)

A collection of vocabulary flashcards covering key concepts from the lecture notes on unions/intersections, set notation, logarithms, inequalities, exponents, and linear programming.

Union

The set of elements that belong to A or B (or both); denoted A ∪ B.

Intersection

The set of elements common to both A and B; denoted A ∩ B.

Interval notation

A notation to describe a set of real numbers using brackets/parentheses, e.g., [-2, 8).

Set description

A verbal description of the elements in a set (e.g., all x such that -3 ≤ x ≤ 4).

Rationalize

To eliminate radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable expression.

Feasible region

The set of all points that satisfy all constraints in a linear programming problem.

Objective function

The function to be maximized or minimized in a linear programming problem (e.g., profit = 30x + 25y).

Constraints

Inequalities that limit the values of the decision variables in an optimization problem.

Graphical method

Solving linear programming problems by graphing constraint lines and identifying the feasible region and optimum.

Inequality

A statement that compares two expressions using

Linear inequality

An inequality of the form ax + by ≤ c (or ≥ c).

Quadratic inequality

An inequality that involves a squared term, such as x^2 - 3x - 4 ≤ 0.

Absolute value

The distance from zero on the number line; |x|, leading to piecewise conditions like |x| < a or |x| > a.

Interval

A connected subset of real numbers, such as (a, b), [a, b], (−∞, b], etc.

Real numbers

All rational and irrational numbers; the entire number line concept.

Rational numbers

Numbers that can be expressed as p/q with integers p, q and q ≠ 0.

Integers

Whole numbers including negatives, zero, and positives (…−2, −1, 0, 1, 2, …).

Natural numbers

Positive integers used for counting (often starting at 1; some definitions include 0).

Prime numbers

Natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and itself.

Logarithm

The inverse operation of exponentiation; log_b(x) = y means b^y = x.

Base (of a logarithm)

The number b in log_b(x) indicating the exponential base.

Log rules (product)

logb(xy) = logb(x) + log_b(y).

Log rules (quotient)

logb(x/y) = logb(x) − log_b(y).

Log rules (power)

logb(x^k) = k logb(x).

Exponent rules (product of powers)

a^m * a^n = a^(m+n).

Exponent rules (power of a power)

(a^m)^n = a^(mn).

Square root

The nonnegative number that, when squared, gives the original number; denoted √.

Cube root

The real number that cubed gives the original number; denoted ∛.

Log base 2

A logarithm with base 2, written as log_2(x).

Boundary (in LP/feasible region)

The edges/lines that bound the feasible region where constraints are active.

Domain

The set of input values for which a function is defined.

Range

The set of possible output values of a function.