Essential Study Notes: Sets, Venn Diagrams, Number Systems, and Unit Analysis for Liberal Arts Math

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Sets and Set Notation

Definition of a Set

A set is a collection of objects, called members or elements of the set. Sets are typically written by listing their members within a pair of braces, { }. If there are too many members to list, three dots (...) are used to indicate a continuing pattern.

Example: The set of natural numbers less than 5 is written as {1, 2, 3, 4}.
Notation: If x is a member of set A, we write x ∈ A.

$Definition of a set$

Set Relationships and Venn Diagrams

Basic Set Relationships

Two sets, A and B, can be related in three fundamental ways, often illustrated using Venn diagrams:

Subset: A is a subset of B if every member of A is also a member of B. The Venn diagram shows A entirely within B.
Disjoint: A and B are disjoint if they have no members in common. Their Venn diagram shows two non-overlapping circles.
Overlapping: A and B are overlapping if they share some members. Their Venn diagram shows two circles that intersect.

$Venn diagram: subset$ $Venn diagrams: disjoint and overlapping sets$

Venn Diagram Applications

Venn diagrams are used to visually represent relationships between sets and to solve problems involving classification and counting.

Example: The Venn diagram below shows the distribution of people at a party by gender and age group.

$Venn diagram: men and under age 30 at a party$

Example: The Venn diagram below shows the distribution of people at a conference by gender, employment status, and education.

$Venn diagram: women, college degree, currently employed at a conference$

The Real Number System

Hierarchy of Number Sets

The real number system is organized into several nested subsets:

Natural numbers: {1, 2, 3, ...}
Whole numbers: {0, 1, 2, 3, ...}
Integers: {..., -2, -1, 0, 1, 2, ...}
Rational numbers: Numbers that can be written as a fraction of two integers.
Irrational numbers: Numbers that cannot be written as a fraction of two integers (e.g., √2, π).
Real numbers: All rational and irrational numbers.

$Diagram of the real number system$

Logic and Categorical Propositions

The Four Standard Categorical Propositions

Categorical propositions are statements about the relationship between two sets, typically expressed in four standard forms:

Form	Example	Subject Set (S)	Predicate Set (P)
All S are P.	All whales are mammals.	whales	mammals
No S are P.	No fish are mammals.	fish	mammals
Some S are P.	Some doctors are women.	doctors	women
Some S are not P.	Some teachers are not men.	teachers	men

$Venn diagrams for categorical propositions: all S are P, no S are P$ $Venn diagrams for categorical propositions: some S are P, some S are not P$

Unit Analysis and Measurement

Definitions and Principles

Unit analysis is the process of working with units to help solve problems. The units of a quantity describe what that quantity measures or counts.

You generally cannot add or subtract numbers with different units.
You can combine different units through multiplication, division, or raising to powers.
Always perform all operations on both the numbers and their associated units.

$Summary of unit analysis$

Key Words and Operations

Key Word or Symbol	Operation	Example
per	Division	miles per hour
of or hyphen	Multiplication	kilowatt-hours
square	Raising to second power	ft × ft = ft2
cube or cubic	Raising to third power	ft × ft × ft = ft3

$Table of unit operations$

Unit Conversions and Measurement Systems

When converting units, multiply by the appropriate conversion factor.
Check that your final answer has the expected units.

$Table of USCS-Metric conversions$

Square and Cubic Units

When converting between square or cubic units, remember to square or cube the conversion factor as appropriate.

Example: 1 yard = 3 feet, so 1 square yard = 9 square feet.

$Diagram showing 1 square yard equals 9 square feet$

Metric System and Prefixes

The metric system uses base units such as the meter (m), kilogram (kg), second (s), and liter (L).
Prefixes indicate multiples or fractions of these units (e.g., kilo-, centi-, milli-).

$Table of common metric prefixes$

Temperature Scales and Conversions

There are three main temperature scales: Fahrenheit (°F), Celsius (°C), and Kelvin (K). Conversion between these scales uses specific formulas:

Celsius to Fahrenheit:
Fahrenheit to Celsius:
Celsius to Kelvin:
Kelvin to Celsius:

$Temperature scales: Fahrenheit, Celsius, Kelvin$

Additional info:

Unit analysis is essential for solving real-world problems involving measurement, conversions, and calculations in science and daily life.
Venn diagrams are a powerful tool for visualizing logical relationships and set operations.