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Chapter 7: Problem 56

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

Short Answer

Expert verified
Using Venn diagrams, we have illustrated the associative law of intersections for sets A, B, and C. We first drew and shaded the region representing \( A \cap (B \cap C) \) and then drew and shaded the region representing \((A \cap B) \cap C\). By comparing the two diagrams, we observed that the shaded regions are the same, thus visually proving the associative law as \( A \cap (B \cap C) = (A \cap B) \cap C \).

Step by step solution

01

Draw the Venn diagram for \(A \cap (B \cap C)\)

First, draw three intersecting circles representing the sets A, B, and C. Label the circles as A, B, and C respectively. Shade the region which represents the intersection of sets B and C, and then shade the region which represents the intersection of set A with the previously shaded region. This represents the expression \( A \cap (B \cap C) \).
02

Draw the Venn diagram for \((A \cap B) \cap C\)

Using another set of three intersecting circles representing the sets A, B, and C, shade the region which represents the intersection of sets A and B. Then, shade the region which represents the intersection of set C with the previously shaded region. This represents the expression \((A \cap B) \cap C\).
03

Compare the two Venn diagrams

Observe the two Venn diagrams made in steps 1 and 2. Notice that the shaded regions representing \( A \cap (B \cap C) \) and \((A \cap B) \cap C\) are the same. This illustrates that the associative law of intersections holds for these sets, as the same region is represented in both Venn diagrams. In conclusion, through the Venn diagrams, we have visually proven that the associative law holds for the intersection of sets A, B, and C, as \( A \cap (B \cap C) = (A \cap B) \cap C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Associative Law
The associative law is a fundamental property in mathematics that applies to various operations. In the context of set theory, it refers to how the grouping of sets does not affect the outcome of their intersection. More specifically, for any sets A, B, and C, the associative law states that \( A \cap (B \cap C) = (A \cap B) \cap C \).
This means that when we are interested in finding the common elements of these sets, it does not matter whether we first find the intersection of B and C and then intersect with A, or find the intersection of A and B and then intersect that result with C.
Either way, the result is the same, confirming the associative nature of the operation.
  • Helps simplify complex problems by re-arranging steps.
  • A core principle in performing operations with multiple sets.
Set Intersection
Set intersection is a basic operation in set theory that identifies elements common to multiple sets. Given two or more sets, their intersection is represented by the symbol \( \cap \) and includes only the elements that are present in all the sets involved.
For instance, in the expression \( A \cap (B \cap C) \), we first find the common elements between B and C and then those common with A. This operation highlights how set intersections are crucial for identifying shared or overlapping properties among sets.
  • Useful in data analysis to find common datasets.
  • Can be visually represented using Venn diagrams for better understanding.
Mathematical Proof
Mathematical proofs provide a logical framework to demonstrate the truth of a statement using a series of logical steps or deductions.
The exercise demonstrates a proof by using visual logic with Venn diagrams. By constructing Venn diagrams for both expressions \( A \cap (B \cap C) \) and \( (A \cap B) \cap C \), we can visually affirm that their shaded regions are equivalent.
This type of proof reinforces the validity of the associative law in set intersections by showing that despite rearranging the grouping, the overlapping regions remain unchanged.
  • Helps in understanding complex concepts through visualization.
  • Critical for validating mathematical properties and theories.
Visual Representation
Visual representation, such as through Venn diagrams, is an effective tool for understanding mathematical concepts. Venn diagrams use overlapping circles to visually depict the relationships between different sets.
In the given exercise, using three intersecting circles allows us to display how the sets A, B, and C overlap with each other. Shading the appropriate sections helps us see the intersections and proves concepts like the associative law.
This visual aid makes abstract ideas concrete, providing an intuitive way to grasp the interactions and relationships between multiple sets.
  • Enhances comprehension by transforming abstract concepts into visual formats.
  • Encourages exploration of set operations in an engaging manner.

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