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Chapter 6: Problem 16

If \(n(A)=60, n(B)=20\), and \(n(A \cap B)=1\), find \(n(A \cap B)\).

Short Answer

Expert verified
The number of elements in the intersection of sets A and B, \(n(A \cap B)\), is 1.

Step by step solution

01

Understand the intersection of sets

Intersection of sets refers to the elements that are common to both sets. In this case, we are looking for the elements that are common to both set A and set B. It is denoted by \(A \cap B\).
02

Use the given information

We are given the following information: \(n(A) = 60\): There are 60 elements in set A \(n(B) = 20\): There are 20 elements in set B \(n(A \cap B) = 1\): There is 1 element that is common to both set A and set B
03

Find \(n(A \cap B)\) using the given information

Since we are already given \(n(A \cap B) = 1\), there is 1 element in the intersection of sets A and B. That's it! The number of elements in the intersection of sets A and B, \(n(A \cap B)\), is 1.

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Key Concepts

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

Intersection of Sets
In set theory, an essential concept often encountered is the **intersection of sets**. The intersection is a new set formed by the elements that are common to the sets being compared. For instance, given two sets, A and B, their intersection is denoted as \(A \cap B\). This notation symbolizes the elements that both A and B contain.
  • For example, if set A includes elements {1, 2, 3} and set B includes elements {3, 4, 5}, the intersection \(A \cap B\) would be \{3\}, since 3 is the only element present in both sets.

Understanding the intersection is crucial when determining relationships or similarities between different sets. When the intersection yields no elements, the sets are said to be disjoint, meaning they don't share any common elements.
Elements of a Set
A **set** is a collection of distinct objects, considered as an object in its own right. The items that make up a set are known as its **elements**. Sets are typically denoted with curly brackets, such as {a, b, c}.
  • The nature of elements in a set can vary widely. For example, elements can be numbers, letters, or even other sets.
  • In set theory, each element must be unique within a particular set. This means that in a set, no element is repeated and the order of elements does not matter.

Grasping the idea of elements and their no-repetition rule is important when working with operations like unions, intersections, and differences of sets. Clearly identifying what constitutes an element of a set helps in analyzing how multiple sets relate to each other.
n(A 鈭� B)
The notation **\(n(A \cap B)\)** is employed to express the number of elements in the intersection of two sets A and B. It provides a measure of how many elements the two sets share.
  • For instance, if \(n(A \cap B) = 3\), this tells us that there are three elements common to both A and B.
  • In our example exercise, we saw that \(n(A \cap B) = 1\), meaning that exactly one element is shared between the two sets.

This notation is crucial when analyzing larger intersections involving multiple sets or in scenarios requiring further analysis of shared components, such as solving probability problems or analyzing data sets. It distills the concept of overlap between sets into a singular, quantifiable value.

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