Problem 10 Let $A$ and $B$ be subsets o... [FREE SOLUTION]

91影视

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 7: Problem 10

Let $A$ and $B$ be subsets of a universal set $U$ and suppose $n(U)=200, n(A)=100, n(B)=80$, and $n(A \cap B)=40$. Compute: a. $n\left(A^{c} \cap B\right)$ b. $n\left(B^{c}\right)$. c. $n\left(A^{c} \cap B^{c}\right)$

Short Answer

Expert verified

a. $n\left(A^{c} \cap B\right) = 40$ b. $n\left(B^{c}\right) = 120$ c. $n\left(A^{c} \cap B^{c}\right) = 80$

Step by step solution

Understand the given information

We are given that $A$ and $B$ are subsets of the universal set $U$. The number of elements in each set is as follows: - $n(U) = 200$ - $n(A) = 100$ - $n(B) = 80$ - $n(A \cap B) = 40$

Calculate n($A^{c} \cap B$)

We know that $n(A \cap B) + n(A^{c} \cap B) = n(B)$. Therefore, we can find the number of elements in $A^{c} \cap B$ using these given values. \[n(A^{c} \cap B) = n(B) - n(A \cap B) = 80 - 40 = 40\] So, $n\left(A^{c} \cap B\right) = 40$.

Calculate n($B^{c}$)

We know that $n(U) = n(B \cup B^{c})$ and $n(B \cup B^{c}) = n(B) + n(B^{c})$. Therefore, we can find the number of elements in $B^{c}$ using these given values. \[n(B^{c}) = n(U) - n(B) = 200 - 80 = 120\] So, $n\left(B^{c}\right) = 120$.

Calculate n($A^{c} \cap B^{c}$)

We know that $n(U) = n(A \cup A^{c})$ and $n(A \cup A^{c}) = n(A) + n(A^{c})$. Therefore, we can find the number of elements in $A^{c}$ using these given values. \[n(A^{c}) = n(U) - n(A) = 200 - 100 = 100\] Now, we can apply the principle of inclusion-exclusion to calculate the number of elements in $A^{c} \cap B^{c}$: \[n(A^{c} \cap B^{c}) = n(A^{c}) + n(B^{c}) - n\left((A^{c} \cap B^{c})^{c}\right)\] Note that $(A^{c} \cap B^{c})^{c} = A \cup B$, so we have: \[n(A^{c} \cap B^{c}) = n(A^{c}) + n(B^{c}) - n(A \cup B)\] And by using the formula for the union of two sets, we get, \[n(A^{c} \cap B^{c}) = n(A^{c}) + n(B^{c}) - (n(A) + n(B) - n(A \cap B))\] Now, substitute the given values: \[n(A^{c} \cap B^{c}) = 100 + 120 - (100 + 80 - 40) = 220 - 140 = 80\] So, $n\left(A^{c} \cap B^{c}\right) = 80$.

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Key Concepts

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

Universal Set

In set theory, the **universal set** is the superset that contains all elements of any set being discussed. It acts as a larger whole from which subsets are derived. Any subset, such as set $A$ or set $B$, includes items that are also part of this universal set.

The **notation** for the universal set is usually $U$.
In our example, $U$ is said to have 200 elements, represented by $n(U)=200$.

Understanding the universal set is crucial for determining complements and other subset relationships since it defines the bounds within which these sets operate.

Subsets

A **subset** is a set whose elements are all contained within another set, called the **universal set**.

For example, if $U$ is the universal set and $A$ is a subset, all elements of $A$ are in $U$.
In the problem, both $A$ and $B$ are subsets of $U$, with $n(A)=100$ and $n(B)=80$.

Subsets help in organizing data and forming intersections or complements, which are used to solve various mathematical problems.

Intersection of Sets

The **intersection of sets** involves elements that are common to multiple sets.

For two sets $A$ and $B$, their intersection is denoted as $A \cap B$.
This intersection tells you which elements exist in both sets.

In our example, the intersection $n(A \cap B)=40$ is how many elements are common between sets $A$ and $B$. Intersections are crucial for understanding overlap among sets and are foundational to set operations in mathematics.

Complement of a Set

The **complement of a set** includes all the elements in the universal set $U$ that are not part of a particular subset.

If $A$ is a subset of $U$, the complement of $A$, denoted $A^c$, includes all elements in $U$ minus those in $A$.
Similar logic applies to $B^c$, the complement of set $B$.

In given problems, it is often necessary to compute complements, such as $n(B^c) = 120$ indicating the elements in $U$ but not in $B$. Calculating set complements aids in resolving how subsets differ from the universal set and each other, enhancing how overlaps and distinctions are managed.

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91影视

Let \(A\) and \(B\) be subsets of a universal set \(U\) and suppose \(n(U)=200, n(A)=100, n(B)=80\), and \(n(A \cap B)=40\). Compute: a. \(n\left(A^{c} \cap B\right)\) b. \(n\left(B^{c}\right)\). c. \(n\left(A^{c} \cap B^{c}\right)\)

Short Answer

Step by step solution

Understand the given information

Calculate n(\(A^{c} \cap B\))

Calculate n(\(B^{c}\))

Calculate n(\(A^{c} \cap B^{c}\))

Key Concepts

Universal Set

Subsets

Intersection of Sets

Complement of a Set

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