math final

statement

A declarative sentence that can be judged as either true or false.

negation (~)

A statement that has the opposite truth value of the original statement.

conjunction (∧)

Logical connective that is true only if both component statements are true.

disjunction (∨)

Logical connective that is true if at least one component statement is true.

conditional (→)

A statement of the form 'If p, then q', which is false only when the hypothesis is true and the conclusion is false.

biconditional (↔)

A statement that is true when both component statements have the same truth value.

truth table

A table that shows the truth value of a logical expression for all possible truth values of its variables.

set

A well-defined collection of distinct objects called elements.

element (∈)

An object that belongs to a set.

roster notation

A way to describe a set by listing all its elements, usually in curly braces. like {1, 2, 3}

set-builder notation

A way to describe a set by specifying the properties that its elements must satisfy.

universal set (U)

The set of all elements considered in a specific context.

empty set (∅)

The unique set containing no elements; it is a subset of every set.

subset (⊆)

Set B is a subset of set A if every element in B is also in A.

proper subset (⊂)

Set B is a proper subset of set A if it is a subset of A, but not equal to A.

complement (A')

The set of all elements in the universal set U that are not in set A.

intersection (∩)

The set of elements common to both set A and set B.

union (∪)

The set of all elements that belong to set A or set B, or both.

difference (A - B)

The set of all elements in set A that are not in set B.

Cartesian product (A × B)

The set of all possible ordered pairs (a, b) where a is from set A and b is from set B.

inequalities

Expressions used to describe a range of values for a variable.

distinction between a statement and an opinion

A statement is a declarative sentence that can be objectively judged as true or false, while an opinion is a subjective belief that cannot be proven true or false.

Promise Kept concept

A method used to understand conditional statements ($p \to q$), focusing on when the 'promise' (the conditional statement) is kept or broken (i.e., when $p$ is true and $q$ is false).

Translating between English and symbolic logic

The process of converting English sentences into logical symbols (like $\sim, \land, \lor, \to, \leftrightarrow$) and vice versa, to analyze their truth values and relationships.

Description method (set notation)

A way to describe a set using a verbal description of its elements, often used for very large or infinite sets where listing is impractical.

Natural numbers ($N$)

The set of positive counting numbers: ${1, 2, 3, …}$

Whole numbers ($W$)

The set of natural numbers including zero: ${0, 1, 2, 3, …}$

Integers ($I$)

The set of whole numbers and their opposites: ${…, -2, -1, 0, 1, 2, …}$

Real numbers ($R$)

The set of all numbers that can be represented on a number line, including rational and irrational numbers.

Venn Diagram

A visual representation using circles or other shapes to show relationships between sets and to illustrate set operations.

Classifying numbers

The process of categorizing numbers into different sets such as natural, whole, integer, rational, or real numbers based on their properties.

Rounding decimals

The process of approximating a decimal number to a specific place value (e.g., tenths, hundredths) by considering the digit in the next place value.

Interval notation

A way to express a set of real numbers between two endpoints using parentheses for exclusive and brackets for inclusive intervals.

Integer Operations

Rules for performing addition, subtraction, multiplication, and division with positive and negative integers.

Converting fractions and decimals

The process of changing a number from fractional form to decimal form (divide numerator by denominator) or from decimal form to fractional form (based on place value).

Reducing fractions

The process of simplifying a fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor.

Ratio

A comparison of two quantities, often expressed as a fraction or using a colon (e.g., $a/b$ or $a:b$).

Proportion

An equation stating that two ratios are equal, often used to solve for an unknown quantity through cross-multiplication.

Order of Operations (PEMDAS)

A rule that specifies the sequence for performing mathematical operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Solving formulas

The process of substituting given values into a mathematical formula and performing the operations to find the value of an unknown variable.

Solving equations

The process of finding the value(s) of a variable that make an equation true, typically by isolating the variable using inverse operations.

Percent application problems

Problems that involve calculating percentages in real-world scenarios, such as sales tax, tips, discounts, and interest.

Sales Tax

To find this, multiply the original price by the sales tax rate (expressed as a decimal).

Symbolic Logic

A branch of mathematics and philosophy that employs symbols and formal systems to represent and analyze logical relationships and arguments.

Converting Percent to Decimal

The transformation of a value expressed as a percentage into its equivalent decimal form, reflecting a proportionate part of a whole.

Mixed Number to Improper Fraction Conversion

To do this: multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

Improper Fraction to Mixed Number Conversion

To do this: divide the numerator by the denominator. The whole number result is your new whole number, and the remainder becomes the new numerator over the original denominator.

Converting Decimal to Percent

To convert a decimal to a percent, multiply the decimal by 100 and add the percent symbol (%). This is equivalent to moving the decimal point two places to the right, indicating its proportion out of one hundred.