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Chapter 6: Problem 56

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

Short Answer

Expert verified
To verify the statement \(A \cap(B \cap C)=(A \cap B) \cap C\) using Venn diagrams, follow these steps: 1. Draw 3 intersecting circles for each of the sets A, B, and C. Shade the region inside all three circles representing \(B \cap C\). Then, shade the region in set A that intersects with the previously shaded region (\(A \cap (B \cap C)\)). 2. Draw another set of 3 intersecting circles. Shade the region where set A intersects with set B, representing \(A \cap B\). Next, shade the area where set C intersects with the previously shaded region (\((A \cap B) \cap C\)). 3. Compare the two Venn diagrams side by side. If the shaded regions in both diagrams are identical, it proves that \(A \cap(B \cap C)=(A \cap B) \cap C\).

Step by step solution

01

Drawing the Venn Diagram for \(A \cap(B \cap C)\)

First, draw 3 circles intersecting each other, representing sets A, B, and C. Label each circle accordingly. Shade the region inside all three circles, which represents the intersection (\(B \cap C\)). Now, shade the region in set A that also intersects with the previously shaded region (\(A \cap (B \cap C)\)). Moving forward, our diagrams should depict this visual intersection.
02

Drawing the Venn Diagram for \((A \cap B) \cap C\)

Draw 3 more intersecting circles, again representing sets A, B, and C, and label each circle accordingly. This time, shade the region where set A intersects with set B, representing \(A \cap B\). Next, shade the area where set C intersects with the previously shaded region (\((A \cap B) \cap C\)). Ensure that our diagrams showcase this visual intersection.
03

Comparing both Venn diagrams

Finally, compare the two Venn diagrams side by side. If the shaded regions in both diagrams are identical, it proves that the given statement is accurate as \(A \cap(B \cap C) = (A \cap B) \cap C\). By following these steps, you'll be able to illustrate the given statement using Venn diagrams.

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Key Concepts

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

Set Theory
Set theory is a fundamental part of mathematics. It deals with understanding and relating different collections of objects, called sets. Sets can be anything: numbers, letters, or even other sets. Essentially, a set is a collection of distinct elements.

Understanding sets helps in solving various mathematical problems. Basic operations include union, intersection, and difference.
  • Union (\( A \cup B \)): All elements from both sets.
  • Intersection (\( A \cap B \)): Elements common to both sets.
  • Difference (\( A - B \)): Elements in one set but not in the other.
Set theory forms the basis for topics like probability, logic, and even databases. When learning about sets, it's important to become familiar with these operations to understand how sets interact with each other in both a theoretical and practical sense.
Mathematical Illustration
Venn diagrams are a popular way to illustrate relationships between sets. They are simple to draw and interpret, making them an excellent educational tool.

Venn diagrams typically use circles to represent sets. The overlap between these circles shows the interaction between the sets, such as intersections or unions.
  • Single Circle: Represents one set.
  • Two Overlapping Circles: Shows intersections and unions of two sets.
  • Three Overlapping Circles: Used for more complex relationships, such as \( A \cap (B \cap C) \) or \( (A \cap B) \cap C \).
Venn diagrams are not just for illustrating set operations; they also help in visual learning by turning abstract concepts into something tangible. This makes it easier to compare and analyze different mathematical expressions.
Intersection of Sets
The intersection of sets is represented by the symbol \( \cap \). It denotes elements that are common to all considered sets.

When we explore \( A \cap (B \cap C) \) and \( (A \cap B) \cap C \), these notations look different but represent the same concept due to the associative property of intersection.
  • Associative Property: \( A \cap (B \cap C) = (A \cap B) \cap C \)
  • Real-world Example: If sets are students enrolled in different courses, the intersection might show students taking all specified courses.
Understanding intersections is key to solving problems involving multiple groups or categories, as it allows us to find commonalities. The visual aid of Venn diagrams highlights these intersections, confirming the theoretical outcomes mathematically.

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Most popular questions from this chapter

If \(n(A)=4, n(B)=5\), and \(n(A \cup B)=9\), find \(n(A \cap B)\).

Data released by the Department of Education regarding the rate (percentage) of ninth-grade students who don't graduate showed that, out of 50 states, 12 states had an increase in the dropout rate during the past 2 yr. 15 states had a dropout rate of at least \(30 \%\) during the past 2 yr. 21 states had an increase in the dropout rate and/or a dropout rate of at least \(30 \%\) during the past 2 yr. a. How many states had both a dropout rate of at least \(30 \%\) and an increase in the dropout rate over the 2 -yr period? b. How many states had a dropout rate that was less than \(30 \%\) but that had increased over the 2 -yr period?

a. \(N \cap F\) b. \((D \cup N)^{c}\)

Let \(A=\\{1,2,3\\} .\) Which of the following sets are equa to \(A\) ? a. \(\\{2,1,3\\}\) b. \(\\{3,2,1\\}\) c. \(\\{0,1,2,3\\}\)

Results of a Department of Education survey of SAT test scores in 22 states showed that 10 states had an average composite SAT score of at least 1000 during the past 3 yr. 15 states had an increase of at least 10 points in the average composite SAT score during the past 3 yr. 8 states had both an average composite SAT score of at least 1000 and an increase in the average composite SAT score of at least 10 points during the past 3 yr. a. How many of the 22 states had composite SAT scores of less than 1000 and showed an increase of at least 10 points over the 3 -yr period? b. How many of the 22 states had composite SAT scores of at least 1000 and did not show an increase of at least 10 points over the 3 -yr period?
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