Question

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

The Venn diagrams of the given expressions demonstrate the distributive laws. For part (a) and part (b), the expressions are equivalent as shown by the overlap of areas in the Venn diagrams.

Step-by-step solution

Part a: Distributive law 1

The expression \(C_{1} \cap\left(C_{2} \cup C_{3}\right)\) refers to elements that are in \(C_{1}\) and either \(C_{2}\) or \(C_{3}\), or both. This can be represented on a Venn diagram as the areas of overlap between circle \(C_{1}\) and the union of circles \(C_{2}\) and \(C_{3}\).

Part a: Distributive law 2

The expression \(\left(C_{1} \cap C_{2}\right) \cup\left(C_{1} \cap C_{3}\right)\) refers to elements that are both in \(C_{1}\) and \(C_{2}\), or both in \(C_{1}\) and \(C_{3}\). This can be shown on a Venn diagram as the union of the areas of overlap between circle \(C_{1}\) and circle \(C_{2}\), and between circle \(C_{1}\) and circle \(C_{3}\).

Part a: Comparing the results

On comparison of the Venn diagrams of the two expressions, we see that they represent the same sets, thus demonstrating the distributive law.

Part b: Distributive law 3

The expression \(C_{1} \cup\left(C_{2} \cap C_{3}\right)\) refers to elements that are in \(C_{1}\), or in both \(C_{2}\) and \(C_{3}\). This can be represented on a Venn diagram as the union of circle \(C_{1}\) and the intersection of circles \(C_{2}\) and \(C_{3}\).

Part b: Distributive law 4

The expression \(\left(C_{1} \cup C_{2}\right) \cap\left(C_{1} \cup C_{3}\right)\) refers to elements that are in either \(C_{1}\) or \(C_{2}\), and either \(C_{1}\) or \(C_{3}\). This can be shown on a Venn diagram as the intersection of the union of circle \(C_{1}\) and circle \(C_{2}\), and the union of circle \(C_{1}\) and circle \(C_{3}\).

Part b: Comparing the results

On comparison of the Venn diagrams of the two expressions, we see that they represent the same set, thus demonstrating the distributive law.