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College Algebra · The Ultimate Cheat Sheet · Sample Lesson
De Morgan’s Laws and Algebra Properties – Counting Sets and Understanding Algebra Rules
By Author Jonathan David
De Morgan’s laws explain how complements interact with unions and intersections. Counting formulas help determine how many elements belong to overlapping finite sets. Algebra properties provide the rules that justify rearranging, simplifying, and solving expressions.
This sample connects those ideas with examples so that the formulas can be read and understood rather than simply memorized.
De Morgan’s Laws – Complementing Unions and Intersections
Let $A$ and $B$ be subsets of a universal set $U$. The complement $A^c$ contains the elements of $U$ that do not belong to $A$. De Morgan’s laws state:
The first law says that an element outside the union belongs to neither set. It must be outside $A$ and outside $B$.
The second law says that an element outside the intersection fails to belong to at least one of the sets. It may still belong to the other set.
Example with a Finite Universal Set
Suppose $U=\{1,2,3,4,5\}$, $A=\{1,2,3\}$, and $B=\{3,4\}$. Then:
Both sides of the first law give the same set. Remember that all complements must use the same universal set.
Counting Elements – The Inclusion–Exclusion Principle
For a finite set $A$, the notation $n(A)$ denotes its number of elements, also called its cardinality.
Adding $n(A)$ and $n(B)$ counts elements in the intersection twice. Subtract the overlap once to count each element of the union exactly once:
For example, suppose 18 students study algebra, 12 study physics, and 7 study both. The number studying at least one of the two subjects is:
Rearranging the same formula lets you find an unknown overlap:
Counting the Union of Three Sets
For three finite sets, add the individual counts, subtract the three pairwise overlaps, and add the triple overlap back once:
An element in all three sets is initially counted three times and then subtracted three times. Adding the triple intersection restores its count to one.
Commutative and Associative Properties
For real numbers, the commutative properties allow the order of addition or multiplication to change:
The associative properties allow the grouping to change:
These rules do not apply in the same way to subtraction or division. For instance, $5-2\ne2-5$. Identify the operation before rearranging an expression.
The Distributive Property
Multiplication distributes over addition:
Every term inside the parentheses receives the outside factor. For example:
Reading the same rule in reverse gives factoring: $ab+ac=a(b+c)$.
Identity Elements and Inverses
An identity element leaves a number unchanged under its operation. An inverse combines with a number to produce the identity.
| Property | Rule | Example |
|---|---|---|
| Additive identity | $a+0=a$ | $7+0=7$ |
| Multiplicative identity | $a\cdot1=a$ | $7\cdot1=7$ |
| Additive inverse | $a+(-a)=0$ | $7+(-7)=0$ |
| Multiplicative inverse | $a\cdot a^{-1}=1$, for $a\ne0$ | $7\cdot\frac17=1$ |
Zero has no multiplicative inverse. The nonzero restriction is essential whenever a reciprocal or division is involved.
Zero, Negative Signs, and Cancellation
Multiplication by zero gives zero, while multiplication by $-1$ gives the additive inverse:
Taking the opposite twice returns the original number. Cancellation in equations is a related but distinct idea:
- If $a+c=b+c$, subtracting $c$ from both sides gives $a=b$.
- If $ac=bc$ and $c\ne0$, dividing both sides by $c$ gives $a=b$.
Equality Properties and Substitution
The reflexive property states $a=a$. The symmetric property says that $a=b$ implies $b=a$. The transitive property says that $a=b$ and $b=c$ imply $a=c$.
Substitution allows equal quantities to replace one another wherever the expressions are defined. Together, these properties explain why a written algebra solution can move from one valid statement to the next.
When reviewing a calculation, name the rule supporting each step. That habit connects a reference sheet to the reasoning required in a complete solution.
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