((AUC) – B) UC ACn(BUUC) U A A B B

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Shade the indicated regions of the following Venn diagrams. Show work

### Venn Diagram Analysis

The image contains two Venn diagrams with three sets labeled A, B, and C, contained within a universal set U. Each diagram represents a different set operation.

#### Diagram 1: \(((A \cup C) - B) \cup C\)

This diagram consists of:

- Three intersecting circles labeled A, B, and C.
- The expression \((A \cup C)\) represents the union of sets A and C, covering all elements in either A or C.
- Subtracting B (\((A \cup C) - B\)) means removing the elements that are also in B from this union.
- Finally, the union with C (\(((A \cup C) - B) \cup C\)) adds back any elements exclusively in C.

**Highlighted Region:**

- The areas in A and C that do not overlap with B, along with all elements of C.

#### Diagram 2: \(A^C \cap (B \cup U^C)\)

This diagram consists of:

- Three intersecting circles labeled A, B, and C.
- The complement of A (\(A^C\)) represents all elements not in A.
- The union of B and the complement of C (\(B \cup U^C\)) includes all elements in B and those outside of C.
- The intersection (\(\cap\)) of \(A^C\) with \(B \cup U^C\) represents elements not in A that either are in B or not in C.

**Highlighted Region:**

- The areas outside of A that are within B and the universal set, ensuring none of them overlap with C.

These diagrams visually illustrate how set operations like unions, intersections, and complements can combine or exclude certain sections of the sets involved.
Transcribed Image Text:### Venn Diagram Analysis The image contains two Venn diagrams with three sets labeled A, B, and C, contained within a universal set U. Each diagram represents a different set operation. #### Diagram 1: \(((A \cup C) - B) \cup C\) This diagram consists of: - Three intersecting circles labeled A, B, and C. - The expression \((A \cup C)\) represents the union of sets A and C, covering all elements in either A or C. - Subtracting B (\((A \cup C) - B\)) means removing the elements that are also in B from this union. - Finally, the union with C (\(((A \cup C) - B) \cup C\)) adds back any elements exclusively in C. **Highlighted Region:** - The areas in A and C that do not overlap with B, along with all elements of C. #### Diagram 2: \(A^C \cap (B \cup U^C)\) This diagram consists of: - Three intersecting circles labeled A, B, and C. - The complement of A (\(A^C\)) represents all elements not in A. - The union of B and the complement of C (\(B \cup U^C\)) includes all elements in B and those outside of C. - The intersection (\(\cap\)) of \(A^C\) with \(B \cup U^C\) represents elements not in A that either are in B or not in C. **Highlighted Region:** - The areas outside of A that are within B and the universal set, ensuring none of them overlap with C. These diagrams visually illustrate how set operations like unions, intersections, and complements can combine or exclude certain sections of the sets involved.
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