Question
Compliment of set
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Answered by
GPT-4o mini
The complement of a set refers to the elements that are not in that set, but are in a larger universal set. If you have a set \( A \) and a universal set \( U \), the complement of set \( A \), often denoted as \( A' \) or \( \bar{A} \), consists of all the elements in \( U \) that are not in \( A \).
### Mathematically:
If \( A \subseteq U \), then the complement of \( A \) is defined as:
\[
A' = \{ x \in U \mid x \notin A \}
\]
### Example:
Let's say the universal set \( U = \{ 1, 2, 3, 4, 5, 6 \} \) and the set \( A = \{ 2, 4, 6 \} \). The complement of set \( A \) would be:
\[
A' = \{ x \in U \mid x \notin A \} = \{ 1, 3, 5 \}
\]
### Properties:
1. **Double Complement**: The complement of the complement of a set returns the original set, i.e., \( (A')' = A \).
2. **Union of a Set and Its Complement**: The union of a set and its complement equals the universal set, i.e., \( A \cup A' = U \).
3. **Intersection of a Set and Its Complement**: The intersection of a set and its complement is the empty set, i.e., \( A \cap A' = \emptyset \).
Understanding the complement of a set is an essential concept in set theory and is widely used in mathematics, logic, and computer science. Let me know if you need more examples or details on related topics!
### Mathematically:
If \( A \subseteq U \), then the complement of \( A \) is defined as:
\[
A' = \{ x \in U \mid x \notin A \}
\]
### Example:
Let's say the universal set \( U = \{ 1, 2, 3, 4, 5, 6 \} \) and the set \( A = \{ 2, 4, 6 \} \). The complement of set \( A \) would be:
\[
A' = \{ x \in U \mid x \notin A \} = \{ 1, 3, 5 \}
\]
### Properties:
1. **Double Complement**: The complement of the complement of a set returns the original set, i.e., \( (A')' = A \).
2. **Union of a Set and Its Complement**: The union of a set and its complement equals the universal set, i.e., \( A \cup A' = U \).
3. **Intersection of a Set and Its Complement**: The intersection of a set and its complement is the empty set, i.e., \( A \cap A' = \emptyset \).
Understanding the complement of a set is an essential concept in set theory and is widely used in mathematics, logic, and computer science. Let me know if you need more examples or details on related topics!