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Chapter 8: Problem 35

Find each indicated intersection or union. \(\\{5,9,11\\} \cap\\{9,11,18\\}\)

Short Answer

Expert verified
\(\{9, 11\}\)

Step by step solution

01

Identify the sets

First, identify the two sets involved in the operation. The first set is \(\{5, 9, 11\}\), and the second set is \(\{9, 11, 18\}\).
02

Understand the operation

The symbol \(\cap\) represents the intersection of two sets. The intersection includes all elements that are common to both sets.
03

Find the common elements

Compare the elements of both sets to find the common elements. Here, the common elements between \(\{5, 9, 11\}\) and \(\{9, 11, 18\}\) are 9 and 11.
04

Write the intersection

The intersection of the two sets is \(\{9, 11\}\). This is because 9 and 11 are the only elements that appear in both sets.

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Key Concepts

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

intersection
The intersection of two sets refers to the elements that are common to both sets. When we talk about intersection, we are interested in finding which elements two or more sets share. The symbol for intersection is \(\cap\). To understand it better, think of it as what both groups have in common.
For example, in the problem given, we have two sets: \(\{5, 9, 11\}\) and \(\{9, 11, 18\}\). The common elements are what we are looking for. By comparing both sets, we identify that the elements 9 and 11 appear in each. Thus, the intersection of these two sets is \(\{9, 11\}\).
sets
A set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are usually denoted by curly brackets \({ }\) and can contain numbers, symbols, or even other sets. For example, \(\{5, 9, 11\}\) is a set with the elements 5, 9, and 11.
There are some characteristics of sets that are important to understand:
  • Each element in a set is unique, meaning no duplicates.
  • The order of elements in a set does not matter, so \(\{5, 9, 11\}\) is the same as \(\{11, 9, 5\}\).

By knowing how to define and recognize sets, you can tackle problems involving sets more easily. In the given exercise, identifying the two sets was a crucial step to find their intersection.
common elements
When we talk about the intersection of sets, we're essentially searching for the common elements. These are the items that show up in both (or all) of the sets we are comparing.
Finding common elements is straightforward:
  • Write down all elements of the first set.
  • Compare each element from the first set with those in the second set.
  • Identify the elements that appear in both sets.

In our example, the sets were \(\{5, 9, 11\}\) and \(\{9, 11, 18\}\). By comparing these, we saw that the elements 9 and 11 are common to both sets. Hence, they are the intersection of the two sets.
Understanding how to find common elements is a fundamental skill in set theory and helps in various areas of mathematics.

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