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Magazine Chapter 2: Problem 9
Use Venn diagrams to verify the following two relationships for any events \(A\) and \(B\) (these are called De Morgan's laws): a. \((A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}\) b. \((A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}\)
Short Answer
Expert verified
Both relationships are verified using Venn diagrams.
Step by step solution
01
Understand De Morgan's laws
De Morgan's laws state how the complement of combined events can be expressed using individual complements. Specifically,
- For union: The complement of the union of two sets is equal to the intersection of their complements.
- For intersection: The complement of the intersection of two sets is equal to the union of their complements.
02
Interpret using Venn Diagrams
A Venn diagram is a pictorial representation of sets and their relationships. We'll use two circles to represent events \(A\) and \(B\), and shade the required areas based on the problem statement to verify De Morgan's laws.
03
Verify (A 鈭� B)鈥� = A鈥� 鈭� B鈥� with Venn Diagrams
- Draw two overlapping circles for events \(A\) and \(B\).- Shade the area that represents \(A \, \cup \, B\) (the union of both circles).- The complement \((A \, \cup \, B)^{\prime}\) is the area outside this shaded region, representing what is not in either circle.- Now, shade the areas that do not belong to \(A\) or \(B\). This shaded area represents both \(A'\) and \(B'\), and thus their intersection, \(A' \, \cap \, B'\).
04
Verify (A 鈭� B)鈥� = A鈥� 鈭� B鈥� with Venn Diagrams
- Using the same circles, shade the area representing \(A \, \cap \, B\) (the overlapping segment of the circles).- The complement \((A \, \cap \, B)^{\prime}\) is the area outside of this overlap, representing everything not in the intersection.- Shade the complete areas for \(A'\) and \(B'\), as their union would cover everything outside \(A \, \cap \, B\).
05
Conclusion
From the Venn diagrams:- For union, the shaded areas match, confirming \((A \, \cup \, B)^{\prime} = A' \, \cap \, B'\).- For intersection, the shaded areas also match, confirming \((A \, \cap \, B)^{\prime} = A' \, \cup \, B'\). De Morgan's laws hold true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
De Morgan's Laws
De Morgan's laws are crucial rules in set theory that connect the concepts of set operations and their complements. They are named after Augustus De Morgan, a British mathematician who formulated these principles. These laws consist of two main principles:
- The complement of the union of two sets is equal to the intersection of their complements.
- The complement of the intersection of two sets is equal to the union of their complements.
- For any sets \(A\) and \(B\): \[ (A \cup B)^{\prime} = A^{\prime} \cap B^{\prime} \]
- For any sets \(A\) and \(B\): \[ (A \cap B)^{\prime} = A^{\prime} \cup B^{\prime} \]
Venn Diagrams
Venn diagrams are a visual tool that helps us understand the relationships between different sets. Named after John Venn, an English logician, these diagrams commonly use circles to demonstrate these relationships. Each circle in the diagram represents a set, and areas where the circles overlap indicate common elements among the sets.For example, if you have two sets, \(A\) and \(B\), portions of the circles will overlap, showing the intersection or common elements. The areas within each circle individually display the elements unique to each set, and the area outside the circles represents the complement of the sets, or elements not in the sets at all.Using Venn diagrams, one can easily visualize how De Morgan's laws apply:
- For the law \((A \cup B)^{\prime} = A^{\prime} \cap B^{\prime}\), the unshaded area outside both circles (neither A nor B) corresponds to the intersection of their complements.
- For the law \((A \cap B)^{\prime} = A^{\prime} \cup B^{\prime}\), the shaded area around the circles shows the union of the complements, encompassing anything not in the intersection.
Set Theory
Set theory is the branch of mathematical logic that studies collections of objects, known as sets. It's the foundation for many areas of mathematics and involves operations such as union, intersection, and complement.
- **Union (\(\cup\))**: Represents all elements that are in either set or both. For sets \(A\) and \(B\), the union is written as \(A \cup B\).
- **Intersection (\(\cap\))**: Represents all elements that are common to both sets. For sets \(A\) and \(B\), the intersection is written as \(A \cap B\).
- **Complement**: Includes all elements not inside a given set. For a set \(A\), its complement is denoted as \(A^{\prime}\).
Event Complements
The concept of complements in set theory is not only fundamental for understanding set operations but also for handling events in probability theory. An event complement refers to all the outcomes that are not part of the observed event. It essentially represents the 'opposite' of the event occurring.If we have an event \(A\), its complement, \(A^{\prime}\), includes all outcomes not in \(A\). For instance, if an event \(A\) is rolling a die and getting a 6, then \(A^{\prime}\) includes all other outcomes: 1, 2, 3, 4, and 5.Complements are particularly useful when working with De Morgan's laws, where they are integral in transforming unions and intersections into more manageable expressions. Recognizing how to efficiently switch between an event and its complement can simplify complex probability problems.
