The Complement of a Set: What's Not Inside?
The Building Blocks: Universal Set, Sets, and Elements
Before we can talk about what is outside a set, we need to define our "universe" of discussion. Think of the universal set as the complete collection of all possible objects or elements relevant to a particular discussion. It is often denoted by the symbol $U$ or $E$ (for the "entire" set).
A set is simply a well-defined collection of distinct objects, which we call elements or members. For example, if our universal set is "All letters in the English alphabet," then a set $A$ could be "The vowels." We write this as $A = \{a, e, i, o, u\}$.
Key Notation:
- Universal Set: $U$
- A set: Capital letter, e.g., $A$, $B$.
- An element: Lowercase letter, e.g., $x$.
- "$\in$" means "is an element of." $a \in A$ reads "a is in set A."
- "$\notin$" means "is not an element of."
Defining and Notating the Complement
Now, the complement of a set $A$ is everything in the universal set $U$ that is not in $A$. It answers the question: "Given everything we are considering, what is left out of this specific group?"
The complement of set $A$ is written with a prime symbol ($A'$), a bar over the letter ($\bar{A}$), or sometimes as $A^c$. We will use $A'$ in this article.
The formal mathematical definition using set-builder notation[2] is:
$A' = \{x \in U \mid x \notin A\}$
This reads: "$A'$ is the set of all elements $x$ in the universal set $U$, such that $x$ is not an element of $A$."
If the universal set is $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{2, 4, 6, 8, 10\}$ (the even numbers in $U$), then the complement $A'$ is $\{1, 3, 5, 7, 9\}$ (the odd numbers in $U$).
Visualizing Complements with Venn Diagrams
Venn diagrams[3] are incredibly useful for picturing sets and their complements. In a Venn diagram, the universal set is represented by a rectangle. Any set within the universal set is drawn as a circle inside this rectangle.
The complement of a set is then shown as the area outside the circle but still inside the rectangle. This shaded "outside" region visually represents all the elements not in the original set.
| Description | Set Notation Example | Venn Diagram Area |
|---|---|---|
| Universal Set ($U$) | $U = \{a, b, c, d, e, f, g\}$ | The entire rectangle |
| Set $A$ | $A = \{a, b, c, d\}$ | The circle labeled A |
| Complement of $A$ ($A'$) | $A' = \{e, f, g\}$ | The shaded area inside the rectangle but outside the circle |
Key Properties and Rules of Complements
The complement operation follows specific logical rules that are consistent and powerful.
Formula: Fundamental Complement Rules
Let $U$ be the universal set and $A$ be any subset of $U$.
- Double Complement Law: The complement of the complement is the original set. $(A')' = A$
- Complement of the Universal Set: The universal set contains everything, so its complement has nothing. $U' = \emptyset$
- Complement of the Empty Set[4]: The empty set has nothing, so its complement is everything. $\emptyset' = U$
- Union with Complement: A set combined with its complement gives the whole universal set. $A \cup A' = U$
- Intersection with Complement: A set and its complement have no common elements. $A \cap A' = \emptyset$
These rules form the basis of De Morgan's Laws[5], which describe how complements interact with unions and intersections of sets:
$(A \cup B)' = A' \cap B'$
The complement of a union is the intersection of the complements.
$(A \cap B)' = A' \cup B'$
The complement of an intersection is the union of the complements.
Applying Complements: From Probability to Real-Life Sorting
The concept of a complement is not just abstract math; it has powerful practical applications. One of the most common is in probability.
In probability, the chance of an event happening plus the chance of it not happening always equals 1 (or 100%). If event $A$ is "it will rain today," then the complement event $A'$ is "it will not rain today." If the probability of rain, $P(A)$, is 30%, then the probability of no rain, $P(A')$, is 70%. This is because $P(A) + P(A') = 1$.
Let's consider a more detailed example. Imagine you are sorting a deck of standard playing cards.
- Universal Set $U$: All 52 cards.
- Set $H$: All Hearts. $H$ contains 13 cards.
- Complement $H'$: All cards that are not Hearts. How many cards are in $H'$? Since there are 52 total and 13 are Hearts, the complement contains $52 - 13 = 39$ cards. This includes all Clubs, Diamonds, and Spades.
This "subtraction from the whole" is the essence of calculating a complement: $n(A') = n(U) - n(A)$, where $n(X)$ means the number of elements in set $X$.
Important Questions About Complements
Q1: Can the complement of a set be empty?
Yes, but only in one specific case. The complement of a set is empty if and only if the original set is the universal set itself. If $A = U$, then there are no elements outside $A$ within $U$, so $A' = \emptyset$.
Q2: Why is defining a universal set so important when talking about complements?
The complement is relative. It depends entirely on what you declare as the universal set. For example, if your set is $A = \{1, 2, 3\}$, its complement changes based on $U$.
- If $U = \{1, 2, 3, 4, 5\}$, then $A' = \{4, 5\}$.
- If $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, then $A' = \{4, 5, 6, 7, 8, 9, 10\}$.
Without a clear universal set, the term "complement" is meaningless.
Q3: What is the relationship between set difference and complement?
The complement is a special case of set difference. The difference between two sets, $A - B$, contains elements in $A$ that are not in $B$. The complement of $A$ is essentially the set difference of the universal set and $A$: $A' = U - A$. So, complement is "subtracting from the universe."
Footnote
[1] Set Theory: The branch of mathematical logic that studies sets, which are collections of objects.
[2] Set-Builder Notation: A mathematical notation for describing a set by stating the properties that its members must satisfy, e.g., $\{x | x \text{ has a certain property}\}$.
[3] Venn Diagram: A diagram using circles or other shapes inside a rectangle (the universal set) to show logical relationships between different sets.
[4] Empty Set (Null Set, $\emptyset$): The unique set that contains no elements.
[5] De Morgan's Laws: Rules in set theory and logic that relate unions and intersections of sets via complementation. Named after mathematician Augustus De Morgan.