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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 laws are verified using Venn diagrams, demonstrating the equivalences.
Step by step solution
01
Identify the Problem
We need to use Venn diagrams to verify two relationships for any events \(A\) and \(B\), which are known as De Morgan's Laws. These laws are: (a) \((A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}\) and (b) \((A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}\).
02
Draw Venn Diagram for Event Union
Draw a Venn diagram with two circles representing sets \(A\) and \(B\). Shade the region for \(A \cup B\) which includes all areas covered by either \(A\) or \(B\).
03
Complement of Union
In the same Venn diagram, shade the region outside the \(A \cup B\) region. This shaded area is \((A \cup B)^{\prime}\), which includes all areas that are outside both sets \(A\) and \(B\).
04
Complement of Individual Events
Draw two separate Venn diagrams for \(A'\) and \(B'\). Shade the area outside of \(A\) to represent \(A'\) and outside of \(B\) to represent \(B'\).
05
Intersection of Complements
In a new diagram with sets \(A'\) and \(B'\), shade the overlapping region where both \(A'\) and \(B'\) are shaded. This region represents \(A' \cap B'\).
06
Verification of First Law
Compare the shaded regions from the complements of the union and the intersection of complements diagrams. Verify that the shaded regions for \((A \cup B)^{\prime}\) and \(A' \cap B'\) are identical, confirming De Morgan's first law.
07
Draw Venn Diagram for Event Intersection
Draw a new Venn diagram with two circles representing sets \(A\) and \(B\). Shade the region where the two sets overlap, representing \(A \cap B\).
08
Complement of Intersection
In the same Venn diagram, shade all areas that are not in the overlap of \(A \cap B\). This shaded area is \((A \cap B)^{\prime}\), which includes all areas that are outside the intersection.
09
Union of Complements
Go back to the diagrams for \(A'\) and \(B'\) and shade all areas that are either in \(A'\) or \(B'\). This region represents \(A' \cup B'\).
10
Verification of Second Law
Compare the shaded regions from the complement of the intersection and the union of complements diagrams. Verify that the shaded regions for \((A \cap B)^{\prime}\) and \(A' \cup B'\) are identical, confirming De Morgan's second law.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Venn Diagrams
Venn diagrams are a visual tool used to represent sets and their relationships with each other. They are particularly effective in explaining concepts such as unions, intersections, and complements of events in set theory and probability. A Venn diagram typically consists of one or more circles within a rectangle.
- The rectangle often represents the universal set, containing all possible elements.
- The circles represent individual sets, and the overlaps between circles represent intersections or common elements between sets.
- The areas outside the circles represent the complement of the sets, which includes all elements not in the specified set.
Set Theory
Set theory is the branch of mathematical logic that studies sets, which are collections of objects. These objects can be anything: numbers, letters, or even other sets. In set theory, we use various operations to create new sets from existing ones, including union, intersection, and complement.
- Sets: Typically denoted with capital letters like \(A\), \(B\), and contain elements within curly brackets such as \(A = \{1, 2, 3\}\).
- Subset: A set \(A\) is a subset of set \(B\) if every element of \(A\) is also an element of \(B\).
- Universal Set: The set containing all elements under discussion, often represented as \(U\).
Probability and Statistics
Probability and statistics involve the study and analysis of random events. In this field, sets represent possible outcomes of an experiment or observation. Events are specific outcomes or combinations of outcomes.
- Probability of an Event: It refers to the measure of how likely an event is to occur. For an event \(A\), this is expressed as \(P(A)\).
- Sample Space: The set of all possible outcomes, often represented by \(S\).
- Events: Subsets of the sample space. For example, if flipping a coin, the sample space is \{Heads, Tails\} and the event "getting a head" is \{Heads\}.
Event Complement
In probability and set theory, the complement of an event refers to all outcomes that are not part of the event. If we have a set \(A\), its complement is usually denoted as \(A'\) or \(A^c\).
- Definition: For an event \(A\), the complement \(A'\) consists of all outcomes in the sample space that are not in \(A\).
- Formula: In terms of probability, if \(S\) is the sample space, then \(P(A') = 1 - P(A)\).
Intersection and Union
Intersection and union are fundamental operations in set theory, often utilized in probability to determine common or combined features of events.
- Intersection: Denoted as \(A \cap B\), it represents elements common to both sets \(A\) and \(B\). For instance, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \cap B = \{3\}\).
- Union: Denoted as \(A \cup B\), it includes all elements present in either \(A\), \(B\), or both. With the same \(A\) and \(B\) as before, \(A \cup B = \{1, 2, 3, 4, 5\}\).
