A ∪ B

Definition

The union of two sets, A and B, is the set of all elements that are in either A, B, or both. It represents the combination of the elements from both sets, without any duplicates.

5 Must Know Facts For Your Next Test

1. The union of sets A and B is denoted as $A \cup B$.
2. The union of sets is commutative, meaning $A \cup B = B \cup A$.
3. The union of sets is associative, so $(A \cup B) \cup C = A \cup (B \cup C)$.
4. If $A \cap B = \emptyset$, then $A \cup B = \{x \mid x \in A \text{ or } x \in B\}$.
5. The union of a set with its complement is the universal set, i.e., $A \cup A' = U$.

Review Questions

• Explain the concept of the union of two sets, A and B, and how it differs from the intersection of the same sets.
  • The union of two sets, A and B, represents the set of all elements that are in either set A, set B, or both sets. This means that the union includes all unique elements from both sets, without any duplicates. In contrast, the intersection of sets A and B is the set of all elements that are common to both sets, i.e., the elements that are present in both A and B. The union operation combines the elements from the two sets, while the intersection operation identifies the elements that are shared between the two sets.
• Describe the properties of the union operation, such as commutativity and associativity, and explain how they are useful in working with set operations.
  • The union operation has two important properties: commutativity and associativity. Commutativity means that the order of the sets in the union operation does not matter, so $A \cup B = B \cup A$. Associativity means that the grouping of the sets in a union operation does not affect the result, so $(A \cup B) \cup C = A \cup (B \cup C)$. These properties are useful because they allow you to simplify and manipulate union expressions more easily, which is important when working with set operations and solving problems involving sets.
• Analyze the relationship between the union of a set and its complement, and explain the significance of this relationship in the context of probability.
  • The relationship between the union of a set and its complement is that the union of a set and its complement is equal to the universal set. Mathematically, this can be expressed as $A \cup A' = U$, where $A'$ represents the complement of set $A$ and $U$ represents the universal set. This relationship is significant in the context of probability because the probability of an event and the probability of its complement add up to 1. In other words, the probability of an event occurring or its complement occurring is certainty, which is represented by the universal set.

Related terms

Set: A collection of distinct objects or elements, which can be finite or infinite.

Intersection (A ∩ B): The set of all elements that are common to both sets A and B.

Complement (A'): The set of all elements that are not in set A.