A'

Definition

A' is a set-theoretic operation that represents the complement of a set A. It is the set of all elements that are not members of the original set A, and it is denoted by the symbol A' or sometimes Ac.

5 Must Know Facts For Your Next Test

1. The complement of a set A, denoted as A', is the set of all elements that are not members of A.
2. The complement of a set A is the set of all elements in the universal set that are not in A.
3. The complement of a set A is the set difference between the universal set and the set A.
4. The complement of the complement of a set A is the set A itself, i.e., (A')' = A.
5. The complement of a set A is an important concept in set theory and is widely used in probability theory and other areas of mathematics.

Review Questions

• Explain the relationship between a set A and its complement A'.
  • The complement of a set A, denoted as A', is the set of all elements that are not members of A. In other words, A' is the set of all elements in the universal set that are not contained in A. The complement of a set A and the set A itself are disjoint sets, meaning they have no elements in common. The union of A and A' is the universal set, and the intersection of A and A' is the empty set.
• Describe how the complement of a set is used in probability theory.
  • In probability theory, the complement of an event A, denoted as A', is the set of all outcomes that are not in the event A. The probability of the complement of an event A is given by the formula P(A') = 1 - P(A), where P(A) is the probability of the event A. This relationship is known as the complement rule and is a fundamental concept in probability theory. The complement of an event is often used to calculate the probability of the opposite or complementary event occurring.
• Analyze the properties of the complement operation and how it relates to other set-theoretic operations.
  • The complement operation has several important properties that relate it to other set-theoretic operations: 1. The complement of the complement of a set A is the set A itself, i.e., (A')' = A. 2. The complement of the union of two sets A and B is the intersection of their complements, i.e., (A ∪ B)' = A' ∩ B'. 3. The complement of the intersection of two sets A and B is the union of their complements, i.e., (A ∩ B)' = A' ∪ B'. 4. The complement of the empty set is the universal set, i.e., ∅' = U. These properties demonstrate the fundamental relationship between the complement operation and other set-theoretic operations, such as union and intersection, and how they can be used to simplify and manipulate set expressions.

Related terms

Set: A set is a well-defined collection of distinct objects, which can be finite or infinite.

Complement:

The complement of a set A is the set of all elements that are not members of A, and it is denoted by A' or Ac.

Union:

The union of two sets A and B is the set of all elements that are in A, B, or both, and is denoted by A ∪ B.