Understanding the Complement of a Set in Mathematics | Definition, Examples, and Properties

Complement

In mathematics, the complement of a set is the set of all elements that do not belong to the given set

In mathematics, the complement of a set is the set of all elements that do not belong to the given set. More formally, if A is a set, then the complement of A, denoted by A’, A̅ or Ac, is the set of all elements that are in the universal set but not in A.

For example, let’s consider the universal set U as the set of all integers between -5 and 5 (including both extremes), and A as the set of positive integers less than 4.
A = {1, 2, 3}
The complement of A, denoted as A’ or Ac, would include all the elements that are in U but not in A. So, the complement of A would contain all the integers from -5 to 5 that are not in A:
A’ = {-5, -4, -3, -2, -1, 0, 4, 5}

It is important to note that the complement of a set depends on the universal set being considered. For example, if the universal set was defined as the set of all real numbers, then the complement of A would be much larger, including all the real numbers that are not positive integers less than 4.

Complements have several important properties:
1. Identity Property: The complement of the complement of a set is the set itself. (A’)’ = A
2. Universal Set Property: The complement of the universal set is the empty set. U’ = ∅
3. Empty Set Property: The complement of the empty set is the universal set. ∅’ = U
4. De Morgan’s Laws: The complement of the union of two sets is the intersection of their complements, and the complement of the intersection of two sets is the union of their complements. (A ∪ B)’ = A’ ∩ B’ and (A ∩ B)’ = A’ ∪ B’

Complements are commonly used in set theory, logic, and probability theory to define operations and relationships between sets.

More Answers: