Union (∪)

Definition

The union of two sets, denoted by the symbol ∪, is the set that contains all the elements that are in either or both of the original sets. It represents the combination or joining of two sets into a single set that includes all the unique elements from both sets.

5 Must Know Facts For Your Next Test

1. The union of two sets A and B is denoted as A ∪ B.
2. The union of sets is commutative, meaning A ∪ B = B ∪ A.
3. The union of sets is associative, so (A ∪ B) ∪ C = A ∪ (B ∪ C).
4. The union of a set with the empty set is the original set, A ∪ ∅ = A.
5. The union of a set with its complement is the universal set, A ∪ A' = U.

Review Questions

• Explain the concept of the union of two sets and how it is represented visually in a Venn diagram.
  • The union of two sets, A and B, is the set that contains all the elements that are in either or both of the original sets. In a Venn diagram, the union of sets A and B is represented by the region that includes all the elements that are in either set A, set B, or both sets A and B. The union is denoted by the symbol A ∪ B and is the combination of the two sets into a single set that includes all the unique elements from both.
• Describe how the union of sets relates to the concept of tree diagrams and their use in probability calculations.
  • In the context of tree diagrams, the union of sets is an important concept. Tree diagrams are used to visualize and calculate probabilities of events, where the branches represent different possible outcomes. The union of sets comes into play when calculating the probability of two or more mutually exclusive events occurring, as the probability of the union of these events is the sum of their individual probabilities. Understanding the union of sets and how it is represented in tree diagrams is crucial for correctly calculating probabilities of compound events.
• Analyze how the properties of the union of sets, such as commutativity and associativity, can be leveraged to simplify set operations and probability calculations.
  • The properties of commutativity and associativity of the union of sets can be very useful in simplifying set operations and probability calculations. The commutative property, which states that A ∪ B = B ∪ A, allows you to rearrange the order of sets in a union without changing the result. This can be helpful when working with complex set expressions or probability problems involving multiple events. The associative property, which states that (A ∪ B) ∪ C = A ∪ (B ∪ C), allows you to group the sets in a union in different ways without altering the final result. Understanding and applying these properties can lead to more efficient and accurate solutions when working with unions of sets, especially in the context of tree diagrams and probability calculations.

Related terms

Intersection (∩): The intersection of two sets is the set that contains only the elements that are common to both sets.

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

Disjoint Sets: Two sets are disjoint if they have no elements in common, meaning their intersection is the empty set.