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Chapter 1: Problem 5

By the use of Venn diagrams, in which the space \(\mathcal{C}\) is the set of points enclosed by a rectangle containing the circles \(C_{1}, C_{2}\), and \(C_{3}\), compare the following sets. These laws are called the distributive laws. (a) \(C_{1} \cap\left(C_{2} \cup C_{3}\right)\) and \(\left(C_{1} \cap C_{2}\right) \cup\left(C_{1} \cap C_{3}\right)\). (b) \(C_{1} \cup\left(C_{2} \cap C_{3}\right)\) and \(\left(C_{1} \cup C_{2}\right) \cap\left(C_{1} \cup C_{3}\right)\).

Short Answer

Expert verified
Both relationships are true and are examples of the distributive laws in action. In (a) \(C_{1} \cap(C_{2} \cup C_{3})\) equals to \((C_{1} \cap C_{2}) \cup(C_{1} \cap C_{3})\), and in (b) \(C_{1} \cup(C_{2} \cap C_{3})\) equals to \((C_{1} \cup C_{2}) \cap(C_{1} \cup C_{3})\). This can be verified through Venn diagrams.

Step by step solution

01

Draw and understand the Venn diagrams

Draw three circles that overlap each other within a rectangle. Label the circles as \(C_{1}\), \(C_{2}\), and \(C_{3}\), and the rectangle as \(\mathcal{C}\). Each section of the circles/rectangle represents different intersections of the sets.
02

Compare \(C_{1} \cap(C_{2} \cup C_{3})\) and \((C_{1} \cap C_{2}) \cup(C_{1} \cap C_{3})\)

The left side of the expression represents the intersection of \(C_{1}\) with the union of \(C_{2}\) and \(C_{3}\). On the Venn diagram, this area would be the part of \(C_{1}\) that also is included in \(C_{2}\) or \(C_{3}\). The right side of the expression is the union of the intersection of \(C_{1}\) and \(C_{2}\) with the intersection of \(C_{1}\) and \(C_{3}\). On a Venn diagram, you'd find that those two areas are identical, proving the distributive law.
03

Compare \(C_{1} \cup(C_{2} \cap C_{3})\) and \((C_{1} \cup C_{2}) \cap(C_{1} \cup C_{3})\)

The left side of the expression represents the union of \(C_{1}\) with the intersection of \(C_{2}\) and \(C_{3}\). On the Venn diagram, this includes all of \(C_{1}\) plus the shared area of \(C_{2}\) and \(C_{3}\). The right side says to find the intersection between the union of \(C_{1}\) with \(C_{2}\) and the union of \(C_{1}\) with \(C_{3}\). Visually, this also includes all of \(C_{1}\) and the shared area between \(C_{2}\) and \(C_{3}\). Because the areas are the same, this again demonstrates the law of distribution.

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