Question

6.12 Consider a Venn diagram picturing two events \(A\) and \(B\) that are not disjoint. a. Shade the event \((A \cup B)^{C} .\) On a separate Venn diagram shade the event \(A^{C} \cup B^{C} .\) How are these two events related? b. Shade the event \((A \cap B)^{C} .\) On a separate Venn diagram shade the event \(A^{C} \cup B^{C}\). How are these two events related? (Note: These two relationships together are called DeMorgan's laws.)

Short answer

In both scenarios, the shaded regions representing \((A \cup B)^{C}\) and \(A^{C} \cup B^{C}\), and then \((A \cap B)^{C}\) and \(A^{C} \cup B^{C}\) are the same, confirming DeMorgan's laws.

Step-by-step solution

Understanding the Sets Notation

Start by understanding what the notations mean. The \(\cup\) symbol represents union of two events - it is the set of all elements that are in either A or B or in both. The \(\cap\) symbol represents intersection of two events - it is the set of all elements that are common to both A and B. The \(C\) represents complement of an event - the set of all elements not in the event.

Shading the Venn Diagram for \((A \cup B)^{C}\)

Sketch a Venn diagram with two overlapping circles representing sets A and B respectively. Shade the region outside of both circles. This represents the complement of the union of A and B \((A \cup B)^{C}\), which is the set of all elements not in either A or B.

Shading the Venn Diagram for \(A^{C} \cup B^{C}\)

Sketch a separate Venn diagram just like in step 2. This time, shade the regions outside A and outside B, this represents the union of the complements of A and B \(A^{C} \cup B^{C}\), which is the set of all elements not in A or not in B.

Comparing the Shaded Regions

Compare the shaded regions in step 2 and step 3. If DeMorgan's law is correct, these two shaded regions should be the same.

Shading the Venn Diagram for \((A \cap B)^{C}\)

Sketch a new Venn diagram, this time shade the region that is outside the intersection of circles A and B, i.e., all areas that do not belong to both A and B.

Comparing the Shaded Regions Again

Compare the shaded region in step 5 with the Venn diagram from step 3. If DeMorgan's law is correct, these two shaded regions should also be the same.

Key concepts

Venn diagram

A Venn diagram is a visual tool used to depict the relationship between different sets in a clear and concise manner. To understand how it helps in set theory, imagine two overlapping circles, each representing a different set. The area where the two circles overlap represents the intersection of the two sets, indicating where they have elements in common. Areas in the individual circles that do not overlap symbolize elements that are unique to each set. When it comes to complex set operations, a Venn diagram can be particularly useful. For example, shading areas corresponding to specific set operations helps in visualizing concepts like union, intersection, and set complement, making abstract ideas much more tangible.

In the context of the exercise, using Venn diagrams allows you to understand the relationships put forth by DeMorgan's laws and see them in action.

Set theory

Set theory is the mathematical study of collections of objects, known as sets. It is a fundamental part of modern mathematics and serves as the foundation for various mathematical disciplines. In set theory, the objects in a set are called elements or members, and they can be anything: numbers, symbols, points, etc. The beauty of set theory lies in its ability to describe operations on these collections, helping us understand concepts such as union, intersection, and set complement. It provides precise language and notation to describe complex mathematical ideas, whether in pure mathematics or applied fields like statistics and probability, as demonstrated by the exercise involving events represented by sets.

Complement of a set

The complement of a set contains all the elements that are not in the original set, given a universal set to which both belong. If you have a universal set, typically denoted by the letter U, and a subset A within it, then the complement of A, denoted as \(A^{C}\), would be the set of all elements in U that are not in A.

Illustrating with a Venn diagram, if a circle within a box represents set A, then the area of the box outside the circle is \(A^{C}\). Understanding the complement is crucial for solving various problems in probability and for grasping the logic behind DeMorgan's laws. In the exercise, we use this concept to visualize set operations by shading the appropriate regions in the Venn diagram.

Union and intersection of sets

Union and intersection are two fundamental operations in set theory. The union of two sets, denoted as \(A \cup B\), is a set that contains all elements that are in set A, set B, or both. It combines the members of each set into one larger set without duplicates.

The intersection of two sets, denoted as \(A \cap B\), is the set containing all elements that are common to both sets A and B. This operation narrows down the focus to the shared elements. These operations are graphically illustrated using Venn diagrams, where the union is represented by the combined shaded area of the two sets and the intersection by the overlapping shaded region. In the exercise, understanding these operations allows for the exploration of the relationship between them as specified by DeMorgan's laws.