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Complement of a Set

Complement of a Set

The elements not contained in a given set. The complement of set A is indicated by AC.

 

Venn diagram showing oval labeled A inside a rectangle; the shaded region outside A is labeled A^C, representing the...

 

 

See also

Venn diagrams, complement of an event

Key Formula

AC={x∣x∈U and x∉A}A^C = \{ x \mid x \in U \text{ and } x \notin A \}
Where:
  • ACA^C = The complement of set A (also written as A' or \bar{A})
  • UU = The universal set — the set of all elements under consideration
  • xx = An individual element
  • ∈\in = Means 'is an element of'
  • ∉\notin = Means 'is not an element of'

Worked Example

Problem: Let the universal set be U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and let A = {2, 4, 6, 8}. Find A^C.
Step 1: Write down the universal set and the given set.
U={1,2,3,4,5,6,7,8,9,10},A={2,4,6,8}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, \quad A = \{2, 4, 6, 8\}
Step 2: Identify every element in U that is NOT in A. Go through each element of U: 1 is not in A ✓, 2 is in A ✗, 3 is not in A ✓, and so on.
Not in A:1,3,5,7,9,10\text{Not in } A: 1, 3, 5, 7, 9, 10
Step 3: Collect those elements into a set to form the complement.
AC={1,3,5,7,9,10}A^C = \{1, 3, 5, 7, 9, 10\}
Step 4: Quick check: every element belongs to either A or A^C (but not both), and together they account for all 10 elements in U.
∣A∣+∣AC∣=4+6=10=∣U∣  ✓|A| + |A^C| = 4 + 6 = 10 = |U| \; \checkmark
Answer: A^C = {1, 3, 5, 7, 9, 10}

Another Example

This example explores two edge cases: the complement of the universal set itself (which is empty) and the complement of the empty set (which is the entire universal set). These boundary cases reinforce the definition.

Problem: Let U = {a, b, c, d, e, f, g} and let B = U. Find B^C. Then let C = ∅ (the empty set) and find C^C.
Step 1: Since B equals the universal set, every element of U is already in B. There are no elements left over.
BC={x∣x∈U and x∉U}=∅B^C = \{ x \mid x \in U \text{ and } x \notin U \} = \emptyset
Step 2: Now consider C = ∅. No elements are in C, so every element of U is not in C.
CC={x∣x∈U and x∉∅}C^C = \{ x \mid x \in U \text{ and } x \notin \emptyset \}
Step 3: Because nothing is excluded, the complement of the empty set is the entire universal set.
CC=U={a,b,c,d,e,f,g}C^C = U = \{a, b, c, d, e, f, g\}
Answer: B^C = ∅ and C^C = U = {a, b, c, d, e, f, g}

Frequently Asked Questions

What is the difference between complement and difference of sets?
The complement of A (written A^C) always refers to everything in the universal set U that is not in A. The set difference A − B (or A \ B) refers to elements in A that are not in B, regardless of any universal set. You can think of the complement as a special case of set difference: A^C = U − A.
Does the complement of a set depend on the universal set?
Yes, absolutely. The complement changes when the universal set changes. For example, if A = {1, 2} and U = {1, 2, 3}, then A^C = {3}. But if U = {1, 2, 3, 4, 5}, then A^C = {3, 4, 5}. You must always know what U is before you can find a complement.
What does the complement of a set look like on a Venn diagram?
On a Venn diagram, the universal set is represented by a rectangle, and set A is a circle inside it. The complement A^C is the shaded region inside the rectangle but outside the circle. This visually shows all elements in U that do not belong to A.

Complement of a Set vs. Complement of an Event (Probability)

Complement of a SetComplement of an Event (Probability)
DefinitionAll elements in the universal set U that are not in set AAll outcomes in the sample space that are not in event A
NotationA^C, A', or ĀA^C, A', or Ā (same notation)
Key formulaA^C = U − AP(A^C) = 1 − P(A)
ContextPure set theory — no probabilities involvedProbability — assigns numerical likelihood to outcomes
When to useWhen listing or describing elements not in a given setWhen finding the probability that an event does NOT occur

Why It Matters

You will encounter set complements frequently in probability, where the complement of an event gives you a shortcut: P(A^C) = 1 − P(A). Complements also appear in logic, database queries, and survey problems (e.g., finding how many people are NOT in a particular group). Understanding complements is essential for working with Venn diagrams and De Morgan's laws, both of which are standard topics in algebra and discrete mathematics courses.

Common Mistakes

Mistake: Finding the complement without defining or knowing the universal set.
Correction: The complement has no meaning without a universal set. Always identify U first. The same set A can have completely different complements under different universal sets.
Mistake: Including elements of A in the complement.
Correction: A and A^C never share any elements. By definition, A ∩ A^C = ∅. If an element appears in both your set and your complement, you have made an error — double-check each element against the original set.

Related Terms