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College Algebra · The Ultimate Cheat Sheet · Sample Lesson
Types of Numbers and Set Laws – College Algebra Definitions and Examples
By Author Jonathan David
College algebra uses different number systems and rules for working with sets. Understanding integers, rational numbers, irrational numbers, and complex numbers helps you identify which values a problem allows. Set notation then provides a precise way to describe those values and their relationships.
This sample explains number classifications, properties of equality, subsets, unions, intersections, complements, and common set laws.
Types of Numbers
| Number Type | Definition | Examples |
|---|---|---|
| Integers | Zero and the positive and negative whole numbers. | $-3,\,-2,\,-1,\,0,\,1,\,2,\,3$ |
| Rational numbers | Numbers expressible as $a/b$, where $a,b$ are integers and $b\ne0$. | $\frac12,\,-\frac34,\,5,\,0.25$ |
| Irrational numbers | Real numbers that cannot be expressed as a ratio of integers. | $\sqrt2,\,\pi$ |
| Complex numbers | Numbers of the form $a+bi$, where $a,b\in\mathbb{R}$ and $i^2=-1$. | $3+2i,\,-i,\,5$ |
These categories overlap. Every integer is rational because it can be written with denominator 1. Every real number is also complex because its imaginary part can be zero.
Rational numbers have terminating or eventually repeating decimal expansions. Irrational numbers have decimal expansions that neither terminate nor eventually repeat.
Properties of Equality
Properties of equality justify how we compare quantities and replace expressions during a solution.
- Reflexive: Every quantity equals itself: $a=a$.
- Symmetric: If $a=b$, then $b=a$.
- Transitive: If $a=b$ and $b=c$, then $a=c$.
- Substitution: Equal quantities can replace one another in an expression wherever it is defined.
For example, if $x=4$, substitution gives $2x+1=2(4)+1=9$. Naming the property helps explain why the step is valid.
Equal Sets, Subsets, and Proper Subsets
Two sets are equal when they contain exactly the same elements. Order and repeated listings do not change a set:
The notation $A\subseteq B$ means every element of $A$ also belongs to $B$. It does not require the sets to be equal. Equality follows when inclusion holds in both directions:
A proper subset additionally requires $A\ne B$. For example, $\{1,2\}\subsetneq\{1,2,3\}$. The symbol $\subsetneq$ makes the strict inclusion explicit.
Union and Intersection
The union contains elements belonging to either set, including elements belonging to both. The intersection contains only elements belonging to both.
Let $A=\{1,2,3\}$ and $B=\{3,4,5\}$. Then:
In set-builder notation, the vertical bar means “such that.” The logical symbols $\land$ and $\lor$ can replace the words “and” and “or,” respectively.
The Complement of a Set
A complement depends on a specified universal set $U$. If $A\subseteq U$, the complement $A^c$ contains the elements of $U$ that are not in $A$:
For example, if $U=\{1,2,3,4,5\}$ and $A=\{1,3,5\}$, then $A^c=\{2,4\}$.
With complements taken relative to the same universal set, the following identities hold:
The empty set, $\varnothing$, contains no elements. A set and its complement together cover the universal set, and they share no elements.
Commutative and Associative Set Laws
The commutative laws say that switching the order of the sets does not change their union or intersection:
The associative laws say that changing the grouping does not change the result when using the same operation:
Distributive Set Laws
Union and intersection distribute over one another:
To understand the second identity, consider an element that belongs to $A$ and also belongs to either $B$ or $C$. It must belong to $A\cap B$ or $A\cap C$. Both sides therefore describe the same collection of elements.
Practice these definitions with small sets you can list completely. Compare the elements on both sides of an identity, and explain each operation in words before working with more abstract notation.
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