Complement Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Complement.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The complement of set A relative to a universal set U is the set of all elements in U that do not belong to A, written A^c or A'.
If the universal set is all students in your school and set A is students who wear glasses, then the complement of A is every student who does NOT wear glasses. It is everything outside the circle in a Venn diagram—the NOT operator applied to a set.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Complement depends on the universal set—what's considered 'everything.'
Common stuck point: Always specify the universal set, or complement is ambiguous.
Sense of Study hint: Write down U first, then cross off every element that is in A. Whatever remains is the complement.
Worked Examples
Example 1
easySolution
- 1 The complement A' (also written A^c or \bar{A}) relative to universal set U is defined as A' = \{x \in U : x \notin A\}.
- 2 List elements of U = \{1,2,3,4,5,6,7,8,9,10\} that are not in A = \{2,4,6,8,10\}: remove the even numbers, leaving the odd numbers.
- 3 Therefore A' = \{1,3,5,7,9\}. Verify: |A| + |A'| = 5 + 5 = 10 = |U| ✓.
Answer
Example 2
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
easyRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.