### Understanding Venn Diagrams in Set TheoryIn set theory, a Venn Diagram is a way to illustrate the logical relationships between different sets. We use circles to represent sets, and the relationships between these sets can be visualized by their overlapping areas. Below, we will analyze four specific set operations using three sets $A$, $B$, and $C$.#### Instructions:Let $A$, $B$, and $C$ be events in the sample space $S$. Use Venn Diagrams to shade the areas representing the following events:### a. $ A \cup (A \cap B) $This expression represents the union of set $A$ with the intersection of sets $A$ and $B$. In a Venn Diagram, this would include all elements of set $A$, because $ A \cap B $ is a subset of $A$, and the union of a set with its subset is the set itself.### b. $ (A \cap B) \cup (A \cap B') $Here, $B'$ represents the complement of set $B$, which includes all elements not in $B$. The expression is the union of the intersection of $A$ and $B$ with the intersection of $A$ and $B'$. This effectively includes all elements of $A$, as they will exist either in or outside $B$.### c. $ A \cup (A' \cap B) $This expression is the union of $A$ with the intersection of $A'$ and $B$. Set $A'$ is the complement of $A$, including all elements not in $A$. The intersection $A' \cap B$ includes only those elements that are in $B$ but not in $A$. Therefore, the union will consist of all elements in $A$ and all elements only in $B$.### d. $ (A \cup B) \cap (A \cup C) $This expression represents the intersection of two unions: $A$ union $B$ and $A$ union $C$. Visually on a Venn Diagram, this would include all elements that are either in $A$, $B$, or $C$, but specifically those that overlap in

Given, A, B and C be the three events in the sample space S. The required Venn diagram for the…

Answered: Let A, B and C be events in the sample…

Let A, B and C be events in the sample space S. Use Venn Diagrams to shade the areas representing the following events a. A U (A B) b. (A N B) U (A N B’)

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Concept explainers

Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.

Tree diagram

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.

Question

### Understanding Venn Diagrams in Set Theory

In set theory, a Venn Diagram is a way to illustrate the logical relationships between different sets. We use circles to represent sets, and the relationships between these sets can be visualized by their overlapping areas. Below, we will analyze four specific set operations using three sets $A$, $B$, and $C$.

#### Instructions:
Let $A$, $B$, and $C$ be events in the sample space $S$. Use Venn Diagrams to shade the areas representing the following events:

### a. $ A \cup (A \cap B) $

This expression represents the union of set $A$ with the intersection of sets $A$ and $B$. In a Venn Diagram, this would include all elements of set $A$, because $ A \cap B $ is a subset of $A$, and the union of a set with its subset is the set itself.

### b. $ (A \cap B) \cup (A \cap B') $

Here, $B'$ represents the complement of set $B$, which includes all elements not in $B$. The expression is the union of the intersection of $A$ and $B$ with the intersection of $A$ and $B'$. This effectively includes all elements of $A$, as they will exist either in or outside $B$.

### c. $ A \cup (A' \cap B) $

This expression is the union of $A$ with the intersection of $A'$ and $B$. Set $A'$ is the complement of $A$, including all elements not in $A$. The intersection $A' \cap B$ includes only those elements that are in $B$ but not in $A$. Therefore, the union will consist of all elements in $A$ and all elements only in $B$.

### d. $ (A \cup B) \cap (A \cup C) $

This expression represents the intersection of two unions: $A$ union $B$ and $A$ union $C$. Visually on a Venn Diagram, this would include all elements that are either in $A$, $B$, or $C$, but specifically those that overlap in

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