Math Problem Statement

Given the Venn diagram below with the universal set U and sets A and B, what are the elements of (A ∪ B)'?

Solution

To find the elements of (AB)(A \cup B)', which is the complement of the union of sets AA and BB, we need to follow these steps:

  1. Union of AA and BB (ABA \cup B):

    • The union of two sets includes all elements in either AA, BB, or both.
    • From the Venn diagram:
      • Elements in AA are: {1,3,5,7,9}\{1, 3, 5, 7, 9\}
      • Elements in BB are: {6,7,8,9,10}\{6, 7, 8, 9, 10\}
    • The union AB={1,3,5,6,7,8,9,10}A \cup B = \{1, 3, 5, 6, 7, 8, 9, 10\}
  2. Complement of ABA \cup B ((AB)(A \cup B)'):

    • The complement includes all elements in the universal set UU that are not in ABA \cup B.
    • Based on the Venn diagram, the elements outside sets AA and BB are {2,4}\{2, 4\}.

So, the elements of (AB)(A \cup B)' are {2,4}\{2, 4\}.

Thus, the correct answer is:

  • {2,4}\{2, 4\}

Do you want more details or have any questions?

Here are 5 related questions to deepen understanding:

  1. What is the difference between a union and an intersection of two sets?
  2. How would you find (AB)(A \cap B)' using a similar approach?
  3. Can you explain how the universal set influences set complements?
  4. What happens if the sets AA and BB are disjoint (have no elements in common)?
  5. How does the concept of a complement change if the universal set is redefined?

Tip: When dealing with complements, always ensure you know the universal set's elements, as the complement is based on this.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Union and Complement of Sets
Venn Diagrams

Formulas

A ∪ B (Union of sets A and B)
(A ∪ B)' (Complement of union of sets)

Theorems

Basic Set Theory Laws
Complement Rule in Set Theory

Suitable Grade Level

Grades 8-10