Problem 57 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use Venn diagrams to illustrate each statement. $$ A \cap(B \cup C)=(A \cap B) \cup(A \cap C) $$

Step-by-Step Solution

Answer

First, draw a Venn diagram with three intersecting circles representing sets A, B, and C. To illustrate A ∩ (B ∪ C), shade all regions containing B and C, and focus on the shaded area where A also exists. For (A ∩ B) ∪ (A ∩ C), shade intersections of A with B and A with C separately, then observe the combined shaded regions. Comparing the shaded regions from both steps, you'll see they are the same, illustrating A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Step 1: Draw a Venn diagram with three sets

Draw a Venn diagram with three sets A, B, and C. Make sure to label each part of the Venn diagram, representing the intersections and complements of the sets. The Venn diagram should look like three intersecting circles, with 7 disjoint regions: A only, B only, C only, A∩B, A∩C, B∩C, and A∩B∩C.

Step 2: Illustrate A ∩ (B ∪ C)

In order to illustrate A ∩ (B ∪ C), we need to find the union of B and C, and then find the intersection with A. To find the union of B and C, shade all regions that contain B and C on the Venn diagram. After shading B ∪ C, focus on the areas of that shaded region that also contain A. This region represents the intersection of A with the union of B and C: A ∩ (B ∪ C).

Step 3: Illustrate (A ∩ B) ∪ (A ∩ C)

To illustrate (A ∩ B) ∪ (A ∩ C), we first need to find the intersection of A and B, and the intersection of A and C. In the Venn diagram, shade the region representing the intersection of A and B and separately shade the region representing the intersection of A and C. After shading both regions, the combined shaded areas represent the union of the intersections A ∩ B and A ∩ C: (A ∩ B) ∪ (A ∩ C).

Step 4: Compare the illustrations

Observe the Venn diagram and compare the regions shaded in Steps 2 and 3. The shaded regions should be the same, illustrating that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). If necessary, redraw the Venn diagram and repeat Steps 2 and 3 to better understand the relationship between these sets.

Key Concepts

Set Theory

Set theory is a fundamental branch of mathematical logic that deals with the study of sets, which are collections of objects. Understanding set theory is essential as it forms the basis for various mathematical concepts. Sets can be anything from numbers, alphabets, or even abstract concepts. A set is usually denoted by curly brackets, such as \( \{a, b, c\} \).

In set theory, there's a universal set which includes all the objects under consideration, while its subsets are smaller collections derived from the universal set. One can perform operations like union, intersection, and complement on these sets. Venn diagrams, typically consisting of overlapping circles, are invaluable tools for visually representing these relationships and operations.

Intersection and Union of Sets

The intersection and union are two primary operations within set theory that help describe how sets interact. The intersection of two sets, denoted as \( A \cap B \), consists of elements that are present in both sets. So, if we have set \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), the intersection \( A \cap B \) is \( \{3\} \), since 3 is the only element that appears in both sets.

On the other hand, the union of two sets, represented by \( A \cup B \), includes all the elements that are in either set, without repetition. For sets \( A \) and \( B \) as defined earlier, \( A \cup B \) would be \( \{1, 2, 3, 4, 5\} \).

Understanding the differences and how these operations can be combined, like in the equation \( A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \), is crucial in comprehending complex relationships between multiple sets.

Illustrating Mathematical Equations

Illustrating mathematical equations, especially those involving set operations, can be greatly simplified using Venn diagrams. These diagrams visually depict the logical relationships between a finite number of sets. In the equation \( A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \), a Venn diagram with three overlapping circles can effectively showcase how the regions of an intersection and union compare.

To visualize \( A \cap(B \cup C) \), one fills in the areas shared between set \( A \) and the union of sets \( B \) and \( C \). This helps students see how intersections concentrate on shared elements, while the union expands to include elements from multiple sets.

By constructing such diagrams and shading the appropriate regions, learners can see real-world representations of abstract concepts, making complex equations more tangible and understandable.