Math Problem Statement

The original question submitted by the user as text.

Solution

The image contains a set theory problem in Indonesian. The problem asks for a Venn diagram to represent the universal set UU and the sets AA and BB, and then to solve specific set operations. Here's the provided information:

  • Universal set U={1,2,3,5,6,7,8}U = \{1, 2, 3, 5, 6, 7, 8\}
  • Set A={2,3,5,7}A = \{2, 3, 5, 7\}
  • Set B={1,3,4,7,8}B = \{1, 3, 4, 7, 8\}

You are asked to solve the following set operations:

  1. ABA - B
  2. BAB - A
  3. ABA \cap B
  4. ABA \cup B
  5. ABA \cap B' (complement of B with respect to U)
  6. BAB \cap A' (complement of A with respect to U)

Let me start solving them one by one:

1. ABA - B (Elements in AA but not in BB)

  • AB=AB={2,3,5,7}{1,3,4,7,8}={2,5}A - B = A \setminus B = \{2, 3, 5, 7\} \setminus \{1, 3, 4, 7, 8\} = \{2, 5\}

2. BAB - A (Elements in BB but not in AA)

  • BA=BA={1,3,4,7,8}{2,3,5,7}={1,4,8}B - A = B \setminus A = \{1, 3, 4, 7, 8\} \setminus \{2, 3, 5, 7\} = \{1, 4, 8\}

3. ABA \cap B (Intersection of AA and BB)

  • AB={2,3,5,7}{1,3,4,7,8}={3,7}A \cap B = \{2, 3, 5, 7\} \cap \{1, 3, 4, 7, 8\} = \{3, 7\}

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

  • AB={2,3,5,7}{1,3,4,7,8}={1,2,3,4,5,7,8}A \cup B = \{2, 3, 5, 7\} \cup \{1, 3, 4, 7, 8\} = \{1, 2, 3, 4, 5, 7, 8\}

5. ABA \cap B' (Elements in AA and not in BB, i.e., the complement of BB with respect to UU)

  • B=UB={1,2,3,5,6,7,8}{1,3,4,7,8}={2,5,6}B' = U \setminus B = \{1, 2, 3, 5, 6, 7, 8\} \setminus \{1, 3, 4, 7, 8\} = \{2, 5, 6\}
  • AB={2,3,5,7}{2,5,6}={2,5}A \cap B' = \{2, 3, 5, 7\} \cap \{2, 5, 6\} = \{2, 5\}

6. BAB \cap A' (Elements in BB and not in AA, i.e., the complement of AA with respect to UU)

  • A=UA={1,2,3,5,6,7,8}{2,3,5,7}={1,6,8}A' = U \setminus A = \{1, 2, 3, 5, 6, 7, 8\} \setminus \{2, 3, 5, 7\} = \{1, 6, 8\}
  • BA={1,3,4,7,8}{1,6,8}={1,8}B \cap A' = \{1, 3, 4, 7, 8\} \cap \{1, 6, 8\} = \{1, 8\}

Would you like to see the Venn diagram representation as well?

Here are five related questions that could expand your understanding of this topic:

  1. What is the difference between ABA \cup B and ABA \cap B?
  2. How would the results change if set AA included elements not in the universal set UU?
  3. Can you calculate (AB)(A \cup B)' with respect to UU?
  4. How does the complement of a set affect its intersection with another set?
  5. What does the symmetric difference between AA and BB represent?

Tip: The complement of a set with respect to the universal set contains all the elements of the universal set that are not in the original set. This is a useful concept when working with Venn diagrams and set operations.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Venn Diagrams
Union
Intersection
Complement of a Set

Formulas

A - B (Difference between sets A and B)
A ∪ B (Union of sets A and B)
A ∩ B (Intersection of sets A and B)
Complement of a set A' or B'

Theorems

Basic Set Theory Operations

Suitable Grade Level

Grades 9-11