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Chapter 8: Problem 38

Find each indicated intersection or union. $$ \\{2,5,9,13\\} \cap\\{5,8,10\\} $$

Short Answer

Expert verified
The intersection is \( \{5\} \).

Step by step solution

01

- Understand Intersection

The intersection (\( \cap \)) of two sets is the set of elements that are common to both sets.
02

- List the Elements

List the elements of both sets:\( \{2, 5, 9, 13\} \) and \( \{5, 8, 10\} \).
03

- Find Common Elements

Identify the elements that appear in both sets. The common element between \( \{2, 5, 9, 13\} \) and \( \{5, 8, 10\} \) is 5.
04

- Write the Result

The intersection of \( \{2, 5, 9, 13\} \) and \( \{5, 8, 10\} \) is \( \{5\} \).

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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 important concept is the intersection of sets. The intersection, denoted by the symbol \( \cap \)\, is a set containing all elements that are present in both of the sets being considered. This means that if you have two sets, the intersection is formed by all the items they both share.
Let's consider the sets \( \{2, 5, 9, 13\} \) and \( \{5, 8, 10\} \). The intersection is formed as follows:
  • Write down all elements of both sets.
  • Identify the elements that appear in both sets.
In this case, we see that 5 is the only number common to both sets. Therefore, the intersection is \( \{5\} \). Understanding intersection helps in many areas, including database searches, filtering data, and solving problems in logic and mathematics.
Common Elements
Finding common elements between two sets is a fundamental task in set theory. To uncover these elements, follow these steps:
  • List all the elements of the first set.
  • List all the elements of the second set.
  • Compare both lists to find the elements that repeat in both sets.
For the given example, the sets were \( \{2, 5, 9, 13\} \) and \( \{5, 8, 10\} \). By listing and comparing these sets, you identify that the number 5 is present in both. Therefore, 5 is a common element. Practicing to find common elements enhances one's ability to recognize patterns and make connections in different fields of study.
Elementary Algebra
Elementary algebra often involves manipulating numbers and variables in simple ways to solve problems. In the context of set theory, it includes operations like finding the intersection of sets. The concept of intersection can be introduced as part of elementary algebra.
Here鈥檚 a breakdown of elementary algebra in relation to set intersection:
  • Recognition of common numbers in lists or sets.
  • Use of symbols like \( \cap \) for operations.
  • A practical approach to solve and simplify problems.
Simple exercises, such as finding common elements between sets, build a strong foundation in various algebraic concepts. It enhances logical thinking and problem-solving skills integral to more advanced mathematics.

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